Each arithmetic operation sometimes becomes too cumbersome to record and they try to simplify it. It used to be the same with the addition operation. It was necessary for people to carry out repeated additions of the same type, for example, to calculate the cost of one hundred Persian carpets, the cost of which is 3 gold coins for each. 3+3+3+…+3 = 300. Because of the cumbersomeness, it was invented to reduce the notation to 3 * 100 = 300. In fact, the notation “three times one hundred” means that you need to take one hundred triplets and add them together. Multiplication took root, gained general popularity. But the world does not stand still, and in the Middle Ages it became necessary to carry out repeated multiplication of the same type. I recall an old Indian riddle about a wise man who asked for wheat grains in the following quantity as a reward for the work done: for the first cell of the chessboard he asked for one grain, for the second - two, the third - four, the fifth - eight, and so on. This is how the first multiplication of powers appeared, because the number of grains was equal to two to the power of the cell number. For example, on the last cell there would be 2*2*2*…*2 = 2^63 grains, which is equal to a number 18 characters long, which, in fact, is the meaning of the riddle.

The operation of raising to a power took root quite quickly, and it also quickly became necessary to carry out addition, subtraction, division and multiplication of degrees. The latter is worth considering in more detail. The formulas for adding powers are simple and easy to remember. In addition, it is very easy to understand where they come from if the power operation is replaced by multiplication. But first you need to understand the elementary terminology. The expression a ^ b (read "a to the power of b") means that the number a should be multiplied by itself b times, and "a" is called the base of the degree, and "b" is the exponent. If the bases of the powers are the same, then the formulas are derived quite simply. Specific example: find the value of the expression 2^3 * 2^4. To know what should happen, you should find out the answer on the computer before starting the solution. Entering this expression into any online calculator, search engine, typing "multiplication of powers with different bases and the same" or a mathematical package, the output will be 128. Now let's write this expression: 2^3 = 2*2*2, and 2^4 = 2 *2*2*2. It turns out that 2^3 * 2^4 = 2*2*2*2*2*2*2 = 2^7 = 2^(3+4) . It turns out that the product of powers with the same base is equal to the base raised to a power equal to the sum of the previous two powers.

You might think that this is an accident, but no: any other example can only confirm this rule. Thus, in general, the formula looks like this: a^n * a^m = a^(n+m) . There is also a rule that any number to the zero power is equal to one. Here we should remember the rule of negative powers: a^(-n) = 1 / a^n. That is, if 2^3 = 8, then 2^(-3) = 1/8. Using this rule, we can prove the equality a^0 = 1: a^0 = a^(n-n) = a^n * a^(-n) = a^(n) * 1/a^(n) , a^ (n) can be reduced and remains one. From this, the rule is derived that the quotient of powers with the same base is equal to this base to a degree equal to the quotient of the dividend and divisor: a ^ n: a ^ m \u003d a ^ (n-m) . Example: Simplify the expression 2^3 * 2^5 * 2^(-7) *2^0: 2^(-2) . Multiplication is a commutative operation, so the multiplication exponents must first be added: 2^3 * 2^5 * 2^(-7) *2^0 = 2^(3+5-7+0) = 2^1 =2. Next, you should deal with the division by a negative degree. It is necessary to subtract the divisor exponent from the dividend exponent: 2^1: 2^(-2) = 2^(1-(-2)) = 2^(1+2) = 2^3 = 8. It turns out that the operation of dividing by a negative degree is identical to the operation of multiplication by a similar positive exponent. So the final answer is 8.

There are examples where non-canonical multiplication of powers takes place. Multiplying powers with different bases is very often much more difficult, and sometimes even impossible. Several examples of various possible approaches should be given. Example: simplify the expression 3^7 * 9^(-2) * 81^3 * 243^(-2) * 729. Obviously, there is a multiplication of powers with different bases. But, it should be noted that all bases are different powers of a triple. 9 = 3^2.1 = 3^4.3 = 3^5.9 = 3^6. Using the rule (a^n) ^m = a^(n*m) , you should rewrite the expression in a more convenient form: 3^7 * (3^2) ^(-2) * (3^4) ^3 * ( 3^5) ^(-2) * 3^6 = 3^7 * 3^(-4) * 3^(12) * 3^(-10) * 3^6 = 3^(7-4+12 -10+6) = 3^(11) . Answer: 3^11. In cases where there are different bases, the rule a ^ n * b ^ n = (a * b) ^ n works for equal indicators. For example, 3^3 * 7^3 = 21^3. Otherwise, when there are different bases and indicators, it is impossible to make a full multiplication. Sometimes you can partially simplify or resort to the help of computer technology.

If you need to raise a specific number to a power, you can use . We will now take a closer look at properties of degrees.

Exponential numbers open up great possibilities, they allow us to convert multiplication into addition, and addition is much easier than multiplication.

For example, we need to multiply 16 by 64. The product of multiplying these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. So 16 times 64=4x4x4x4x4 which is also 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now let's use the rule. 16=4 2 , or 2 4 , 64=4 3 , or 2 6 , while 1024=6 4 =4 5 , or 2 10 .

Therefore, our problem can be written in another way: 4 2 x4 3 =4 5 or 2 4 x2 6 =2 10, and each time we get 1024.

We can solve a number of similar examples and see that the multiplication of numbers with powers reduces to addition of exponents, or an exponent, of course, provided that the bases of the factors are equal.

Thus, we can, without multiplying, immediately say that 2 4 x2 2 x2 14 \u003d 2 20.

This rule is also true when dividing numbers with powers, but in this case, e the exponent of the divisor is subtracted from the exponent of the dividend. Thus, 2 5:2 3 =2 2 , which in ordinary numbers is equal to 32:8=4, that is, 2 2 . Let's summarize:

a m x a n \u003d a m + n, a m: a n \u003d a m-n, where m and n are integers.

At first glance, it might seem that multiplication and division of numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16 in this form, that is, 2 3 and 2 4, but how to do this with the numbers 7 and 17? Or what to do in those cases when the number can be represented in exponential form, but the bases of exponential expressions of numbers are very different. For example, 8×9 is 2 3 x 3 2 , in which case we cannot sum the exponents. Neither 2 5 nor 3 5 is the answer, nor is the answer between the two.

Then is it worth bothering with this method at all? Definitely worth it. It provides huge advantages, especially for complex and time-consuming calculations.

Addition and subtraction of powers

Obviously, numbers with powers can be added like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2 .
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds the same powers of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2 .

It is also obvious that if we take two squares a, or three squares a, or five squares a.

But degrees various variables and various degrees identical variables, must be added by adding them to their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3 .

It is obvious that the square of a, and the cube of a, is neither twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6 .

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahend must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Power multiplication

Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3 .

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n is;

And a m , is taken as a factor as many times as the degree m is equal to;

So, powers with the same bases can be multiplied by adding the exponents.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are − negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y-n .y-m = y-n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y).(a + y) = a 2 - y 2 .
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4 .
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8 .

Division of powers

Numbers with powers can be divided like other numbers by subtracting from the divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is a 3 .

Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing powers with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.

Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$

It is necessary to master the multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce exponents in $\frac $ Answer: $\frac $.

2. Reduce the exponents in $\frac$. Answer: $\frac $ or 2x.

3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

degree properties

We remind you that in this lesson we understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.

An exponent with a natural exponent has several important properties that allow you to simplify calculations in exponent examples.

Property #1
Product of powers

When multiplying powers with the same base, the base remains unchanged, and the exponents are added.

a m a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of powers also affects the product of three or more powers.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
• Present as a degree.
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Please note that in the indicated property it was only about multiplying powers with the same bases.. It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5 . This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36 and 3 5 = 243

Property #2
Private degrees

When dividing powers with the same base, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

• Write the quotient as a power
  (2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
• Calculate.

11 3 - 2 4 2 - 1 = 11 4 = 44
Example. Solve the equation. We use the property of partial degrees.
3 8: t = 3 4

Answer: t = 3 4 = 81

Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

Example. Find the value of an expression using degree properties.

2 11 − 5 = 2 6 = 64

Please note that property 2 dealt only with the division of powers with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1 . This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4

Property #3
Exponentiation

When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.

(a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

How to multiply powers

How to multiply powers? Which powers can be multiplied and which cannot? How do you multiply a number by a power?

In algebra, you can find the product of powers in two cases:

1) if the degrees have the same basis;

2) if the degrees have the same indicators.

When multiplying powers with the same base, the base must remain the same, and the exponents must be added:

When multiplying degrees with the same indicators, the total indicator can be taken out of brackets:

Consider how to multiply powers, with specific examples.

The unit in the exponent is not written, but when multiplying the degrees, they take into account:

When multiplying, the number of degrees can be any. It should be remembered that you can not write the multiplication sign before the letter:

In expressions, exponentiation is performed first.

If you need to multiply a number by a power, you must first perform exponentiation, and only then - multiplication:

Multiplying powers with the same base

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In this lesson, we will learn how to multiply powers with the same base. First, we recall the definition of the degree and formulate a theorem on the validity of the equality . Then we give examples of its application to specific numbers and prove it. We will also apply the theorem to solve various problems.

Topic: Degree with a natural indicator and its properties

Lesson: Multiplying powers with the same bases (formula)

1. Basic definitions

Basic definitions:

n- exponent,

— n-th power of a number.

2. Statement of Theorem 1

Theorem 1. For any number a and any natural n and k equality is true:

In other words: if a- any number; n and k natural numbers, then:

Hence rule 1:

3. Explaining tasks

Conclusion: special cases confirmed the correctness of Theorem No. 1. Let us prove it in the general case, that is, for any a and any natural n and k.

4. Proof of Theorem 1

Given a number a- any; numbers n and k- natural. Prove:

The proof is based on the definition of the degree.

5. Solution of examples using Theorem 1

Example 1: Present as a degree.

To solve the following examples, we use Theorem 1.

g)

6. Generalization of Theorem 1

Here is a generalization:

7. Solution of examples using a generalization of Theorem 1

8. Solving various problems using Theorem 1

Example 2: Calculate (you can use the table of basic degrees).

a) (according to the table)

b)

Example 3: Write as a power with base 2.

a)

Example 4: Determine the sign of the number:

, a - negative because the exponent at -13 is odd.

Example 5: Replace ( ) with a power with a base r:

We have , that is .

9. Summing up

1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 7. 6th edition. M.: Enlightenment. 2010

1. School Assistant (Source).

1. Express as a degree:

a B C D E)

3. Write as a power with base 2:

4. Determine the sign of the number:

a)

5. Replace ( ) with a power of a number with a base r:

a) r 4 ( ) = r 15 ; b) ( ) r 5 = r 6

Multiplication and division of powers with the same exponents

In this lesson, we will study the multiplication of powers with the same exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising a power to a power. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we will solve a number of typical problems.

Reminder of basic definitions and theorems

Here a- base of degree

— n-th power of a number.

Theorem 1. For any number a and any natural n and k equality is true:

When multiplying powers with the same base, the exponents are added, the base remains unchanged.

Theorem 2. For any number a and any natural n and k, such that n > k equality is true:

When dividing powers with the same base, the exponents are subtracted, and the base remains unchanged.

Theorem 3. For any number a and any natural n and k equality is true:

All the above theorems were about powers with the same grounds, this lesson will consider degrees with the same indicators.

Examples for multiplying powers with the same exponents

Consider the following examples:

Let's write out the expressions for determining the degree.

Conclusion: From the examples, you can see that , but this still needs to be proven. We formulate the theorem and prove it in the general case, that is, for any a and b and any natural n.

Statement and proof of Theorem 4

For any numbers a and b and any natural n equality is true:

Proof Theorem 4 .

By definition of degree:

So we have proven that .

To multiply powers with the same exponent, it is enough to multiply the bases, and leave the exponent unchanged.

Statement and proof of Theorem 5

We formulate a theorem for dividing powers with the same exponents.

For any number a and b() and any natural n equality is true:

Proof Theorem 5 .

Let's write down and by definition of degree:

Statement of theorems in words

So we have proven that .

To divide degrees with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.

Solution of typical problems using Theorem 4

Example 1: Express as a product of powers.

To solve the following examples, we use Theorem 4.

To solve the following example, recall the formulas:

Generalization of Theorem 4

Generalization of Theorem 4:

Solving Examples Using Generalized Theorem 4

Continued solving typical problems

Example 2: Write as a degree of product.

Example 3: Write as a power with an exponent of 2.

Calculation Examples

Example 4: Calculate in the most rational way.

2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF

3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7 .M .: Education. 2006

2. School assistant (Source).

1. Present as a product of powers:

a) ; b) ; in) ; G) ;

2. Write down as the degree of the product:

3. Write in the form of a degree with an indicator of 2:

4. Calculate in the most rational way.

Mathematics lesson on the topic "Multiplication and division of powers"

Sections: Mathematics

Pedagogical goal:

the student will learn to distinguish between the properties of multiplication and division of powers with a natural exponent; apply these properties in the case of the same bases;

the student will have the opportunity be able to perform transformations of degrees with different bases and be able to perform transformations in combined tasks.

Tasks:

organize the work of students by repeating previously studied material;

ensure the level of reproduction by performing exercises of various types;

organize self-assessment of students through testing.

Activity units of the doctrine: determination of the degree with a natural indicator; degree components; definition of private; associative law of multiplication.

I. Organization of a demonstration of mastering the existing knowledge by students. (step 1)

a) Updating knowledge:

2) Formulate a definition of the degree with a natural indicator.

a n \u003d a a a a ... a (n times)

b k \u003d b b b b a ... b (k times) Justify your answer.

II. Organization of self-assessment of the trainee by the degree of possession of relevant experience. (step 2)

Test for self-examination: (individual work in two versions.)

A1) Express the product 7 7 7 7 x x x as a power:

A2) Express as a product the degree (-3) 3 x 2

A3) Calculate: -2 3 2 + 4 5 3

I select the number of tasks in the test in accordance with the preparation of the class level.

For the test, I give a key for self-testing. Criteria: pass-fail.

III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)

calculate: 2 2 2 3 = ? 3 3 3 2 3 =?

Simplify: a 2 a 20 =? b 30 b 10 b 15 = ?

In the course of solving problems 1) and 2), the students propose a solution, and I, as a teacher, organize a class to find a way to simplify the powers when multiplying with the same bases.

Teacher: come up with a way to simplify powers when multiplying with the same base.

An entry appears on the cluster:

The theme of the lesson is formulated. Multiplication of powers.

Teacher: come up with a rule for dividing degrees with the same bases.

Reasoning: what action checks division? a 5: a 3 = ? that a 2 a 3 = a 5

I return to the scheme - a cluster and supplement the entry - ..when dividing, subtract and add the topic of the lesson. ...and division of degrees.

IV. Communication to students of the limits of knowledge (as a minimum and as a maximum).

Teacher: the task of the minimum for today's lesson is to learn how to apply the properties of multiplication and division of powers with the same bases, and the maximum: to apply multiplication and division together.

Write on the board : a m a n = a m + n ; a m: a n = a m-n

V. Organization of the study of new material. (step 5)

a) According to the textbook: No. 403 (a, c, e) tasks with different wording

No. 404 (a, e, f) independent work, then I organize a mutual check, I give the keys.

b) For what value of m does the equality hold? a 16 a m \u003d a 32; x h x 14 = x 28; x 8 (*) = x 14

Task: come up with similar examples for division.

c) No. 417(a), No. 418 (a) Traps for students: x 3 x n \u003d x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 \u003d a 2.

VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, not teachers, to study this topic) (step 6)

diagnostic work.

Test(place the keys on the back of the test).

Task options: present as a degree the quotient x 15: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 true; find the value of the expression h 0: h 2 with h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .

Summary of the lesson. Reflection. I divide the class into two groups.

Find the arguments of group I: in favor of knowledge of the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers, draw conclusions. In subsequent lessons, you can offer statistical data and name the rubric “It doesn’t fit in my head!”

The average person eats 32 10 2 kg of cucumbers during their lifetime.

The wasp is capable of making a non-stop flight of 3.2 10 2 km.

When glass cracks, the crack propagates at a speed of about 5 10 3 km/h.

A frog eats over 3 tons of mosquitoes in its lifetime. Using the degree, write in kg.

The most prolific is the ocean fish - the moon (Mola mola), which lays up to 300,000,000 eggs with a diameter of about 1.3 mm in one spawning. Write this number using a degree.

VII. Homework.

History reference. What numbers are called Fermat numbers.

P.19. #403, #408, #417

Used Books:

Textbook "Algebra-7", authors Yu.N. Makarychev, N.G. Mindyuk and others.

Didactic material for grade 7, L.V. Kuznetsova, L.I. Zvavich, S.B. Suvorov.

Encyclopedia of Mathematics.

Journal "Quantum".

Properties of degrees, formulations, proofs, examples.

After the degree of the number is determined, it is logical to talk about degree properties. In this article, we will give the basic properties of the degree of a number, while touching on all possible exponents. Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples.

Page navigation.

Properties of degrees with natural indicators

By definition of a power with a natural exponent, the power of a n is the product of n factors, each of which is equal to a . Based on this definition, and using real number multiplication properties, we can obtain and justify the following properties of degree with natural exponent:

the main property of the degree a m ·a n =a m+n , its generalization a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k ;

the property of partial powers with the same bases a m:a n =a m−n ;

product degree property (a b) n =a n b n , its extension (a 1 a 2 a k) n = a 1 n a 2 n a k n ;

quotient property in kind (a:b) n =a n:b n ;

exponentiation (a m) n =a m n , its generalization (((a n 1) n 2) ...) n k =a n 1 ·n 2 ·... n k ;

comparing degree with zero:

• if a>0 , then a n >0 for any natural n ;
• if a=0 , then a n =0 ;
• if a 2 m >0 , if a 2 m−1 n ;
• if m and n are natural numbers such that m>n , then for 0m n , and for a>0 the inequality a m >a n is true.

We immediately note that all the written equalities are identical under the specified conditions, and their right and left parts can be interchanged. For example, the main property of the fraction a m a n = a m + n with simplification of expressions often used in the form a m+n = a m a n .

Now let's look at each of them in detail.

Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.

Let us prove the main property of the degree. By definition of a degree with a natural exponent, the product of powers with the same bases of the form a m a n can be written as the product . Due to the properties of multiplication, the resulting expression can be written as , and this product is the power of a with natural exponent m+n , that is, a m+n . This completes the proof.

Let us give an example that confirms the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, according to the main property of the degree, we can write the equality 2 2 ·2 3 =2 2+3 =2 5 . Let's check its validity, for which we calculate the values ​​of the expressions 2 2 ·2 3 and 2 5 . Performing exponentiation, we have 2 2 2 3 =(2 2) (2 2 2)=4 8=32 and 2 5 =2 2 2 2 2=32 , since we get equal values, then the equality 2 2 2 3 =2 5 is true, and it confirms the main property of the degree.

The main property of a degree based on the properties of multiplication can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 a n 2 a n k =a n 1 +n 2 +…+n k is true.

For example, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 = (2.1) 3+3+4+7 =(2.1) 17 .

You can move on to the next property of degrees with a natural indicator - the property of partial powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n , the equality a m:a n =a m−n is true.

Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that it is impossible to divide by zero. The condition m>n is introduced so that we do not go beyond natural exponents. Indeed, for m>n, the exponent a m−n is a natural number, otherwise it will be either zero (which happens when m−n) or a negative number (which happens when m m−n a n =a (m−n) + n = a m From the obtained equality a m−n a n = a m and from the relation of multiplication with division it follows that a m−n is a partial power of a m and a n This proves the property of partial powers with the same bases.

Let's take an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the considered property of the degree corresponds to the equality π 5: π 2 = π 5−3 = π 3.

Now consider product degree property: the natural degree n of the product of any two real numbers a and b is equal to the product of the degrees a n and b n , that is, (a b) n =a n b n .

Indeed, by definition of a degree with a natural exponent, we have . The last product, based on the properties of multiplication, can be rewritten as , which is equal to a n b n .

Here's an example: .

This property extends to the degree of the product of three or more factors. That is, the natural degree property n of the product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n .

For clarity, we show this property with an example. For the product of three factors to the power of 7, we have .

The next property is natural property: the quotient of the real numbers a and b , b≠0 to the natural power n is equal to the quotient of the powers a n and b n , that is, (a:b) n =a n:b n .

The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n , and from the equality (a:b) n b n =a n it follows that (a:b) n is a quotient of a n to b n .

Let's write this property using the example of specific numbers: .

Now let's voice exponentiation property: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of a with exponent m·n , that is, (a m) n =a m·n .

For example, (5 2) 3 =5 2 3 =5 6 .

The proof of the power property in a degree is the following chain of equalities: .

The considered property can be extended to degree within degree within degree, and so on. For example, for any natural numbers p, q, r, and s, the equality . For greater clarity, let's give an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10 .

It remains to dwell on the properties of comparing degrees with a natural exponent.

We start by proving the comparison property of zero and power with a natural exponent.

First, let's justify that a n >0 for any a>0 .

The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication allow us to assert that the result of multiplying any number of positive numbers will also be a positive number. And the power of a with natural exponent n is, by definition, the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a the degree of a n is a positive number. By virtue of the proved property 3 5 >0 , (0.00201) 2 >0 and .

It is quite obvious that for any natural n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0 .

Let's move on to negative bases.

Let's start with the case when the exponent is an even number, denote it as 2 m , where m is a natural number. Then . According to the rule of multiplication of negative numbers, each of the products of the form a a is equal to the product of the modules of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive. and degree a 2 m . Here are examples: (−6) 4 >0 , (−2,2) 12 >0 and .

Finally, when the base of a is a negative number and the exponent is an odd number 2 m−1, then . All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3 17 n n is the product of the left and right parts of n true inequalities a properties of inequalities, the inequality being proved is of the form a n n . For example, due to this property, the inequalities 3 7 7 and .

It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of the two degrees with natural indicators and the same positive bases less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree is greater, the indicator of which is greater. We turn to the proof of this property.

Let us prove that for m>n and 0m n . To do this, we write the difference a m − a n and compare it with zero. The written difference after taking a n out of brackets will take the form a n ·(a m−n −1) . The resulting product is negative as the product of a positive number a n and a negative number a m−n −1 (a n is positive as a natural power of a positive number, and the difference a m−n −1 is negative, since m−n>0 due to the initial condition m>n , whence it follows that for 0m−n it is less than one). Therefore, a m − a n m n , which was to be proved. For example, we give the correct inequality.

It remains to prove the second part of the property. Let us prove that for m>n and a>1, a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree of a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1, the degree of a m−n is greater than one . Therefore, a m − a n >0 and a m >a n , which was to be proved. This property is illustrated by the inequality 3 7 >3 2 .

Properties of degrees with integer exponents

Since positive integers are natural numbers, then all properties of powers with positive integer exponents exactly coincide with the properties of powers with natural exponents listed and proven in the previous paragraph.

We defined a degree with a negative integer exponent, as well as a degree with a zero exponent, so that all properties of degrees with natural exponents expressed by equalities remain valid. Therefore, all these properties are valid both for zero exponents and for negative exponents, while, of course, the bases of the degrees are nonzero.

So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true properties of degrees with integer exponents:

• a m a n \u003d a m + n;
• a m: a n = a m−n ;
• (a b) n = a n b n ;
• (a:b) n =a n:b n ;
• (a m) n = a m n ;
• if n is a positive integer, a and b are positive numbers, and a n n and a−n>b−n ;
• if m and n are integers, and m>n , then for 0m n , and for a>1, the inequality a m >a n is satisfied.

For a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.

It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, we need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p q , (a − p) q =a (−p) q , (a p ) −q =a p (−q) and (a −p) −q =a (−p) (−q) . Let's do it.

For positive p and q, the equality (a p) q =a p·q was proved in the previous subsection. If p=0 , then we have (a 0) q =1 q =1 and a 0 q =a 0 =1 , whence (a 0) q =a 0 q . Similarly, if q=0 , then (a p) 0 =1 and a p 0 =a 0 =1 , whence (a p) 0 =a p 0 . If both p=0 and q=0 , then (a 0) 0 =1 0 =1 and a 0 0 =a 0 =1 , whence (a 0) 0 =a 0 0 .

Let us now prove that (a −p) q =a (−p) q . By definition of a degree with a negative integer exponent , then . By the property of the quotient in the degree, we have . Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the multiplication rules, can be written as a (−p) q .

Similarly .

And .

By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.

In the penultimate of the properties written down, it is worth dwelling on the proof of the inequality a −n >b −n , which is true for any negative integer −n and any positive a and b for which the condition a . We write and transform the difference between the left and right parts of this inequality: . Since by condition a n n , therefore, b n − a n >0 . The product a n ·b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as a quotient of positive numbers b n − a n and a n b n . Hence, whence a −n >b −n , which was to be proved.

The last property of degrees with integer exponents is proved in the same way as the analogous property of degrees with natural exponents.

Properties of powers with rational exponents

We defined the degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, degrees with fractional exponents have the same properties as degrees with integer exponents. Namely:

1. property of the product of powers with the same base for a>0 , and if and , then for a≥0 ;
2. property of partial powers with the same bases for a>0 ;
3. fractional product property for a>0 and b>0 , and if and , then for a≥0 and (or) b≥0 ;
4. quotient property to a fractional power for a>0 and b>0 , and if , then for a≥0 and b>0 ;
5. degree property in degree for a>0 , and if and , then for a≥0 ;
6. the property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is valid, and for p p >b p ;
7. the property of comparing powers with rational exponents and equal bases: for rational numbers p and q, p>q for 0p q, and for a>0, the inequality a p >a q .

The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, on the properties of the arithmetic root of the nth degree, and on the properties of a degree with an integer exponent. Let's give proof.

By definition of the degree with a fractional exponent and , then . The properties of the arithmetic root allow us to write the following equalities. Further, using the property of the degree with an integer exponent, we obtain , whence, by the definition of a degree with a fractional exponent, we have , and the exponent of the degree obtained can be converted as follows: . This completes the proof.

The second property of powers with fractional exponents is proved in exactly the same way:


The rest of the equalities are proved by similar principles:


We turn to the proof of the next property. Let us prove that for any positive a and b , a 0 the inequality a p p is valid, and for p p >b p . We write the rational number p as m/n , where m is an integer and n is a natural number. Conditions p 0 in this case will be equivalent to conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of the roots, we have , and since a and b are positive numbers, then, based on the definition of the degree with a fractional exponent, the resulting inequality can be rewritten as , that is, a p p .

Similarly, when m m >b m , whence , that is, and a p >b p .

It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q . We can always reduce the rational numbers p and q to a common denominator, let us get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from the rule for comparing ordinary fractions with the same denominators. Then, by the property of comparing powers with the same bases and natural exponents, for 0m 1 m 2 , and for a>1, the inequality a m 1 >a m 2 . These inequalities in terms of the properties of the roots can be rewritten, respectively, as and . And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively. From here we draw the final conclusion: for p>q and 0p q , and for a>0, the inequality a p >a q .

Properties of degrees with irrational exponents

From how a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0 , b>0 and irrational numbers p and q the following are true properties of degrees with irrational exponents:

1. a p a q = a p + q ;
2. a p:a q = a p−q ;
3. (a b) p = a p b p ;
4. (a:b) p =a p:b p ;
5. (a p) q = a p q ;
6. for any positive numbers a and b , a 0 the inequality a p p is valid, and for p p >b p ;
7. for irrational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q .

From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.

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Lesson on the topic: "Rules for multiplying and dividing powers with the same and different exponents. Examples"

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The purpose of the lesson: learn how to perform operations with powers of a number.

To begin with, let's recall the concept of "power of a number". An expression like $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.

The reverse is also true: $a^n= \underbrace( a * a * \ldots * a )_(n)$.

This equality is called "recording the degree as a product". It will help us determine how to multiply and divide powers.
Remember:
a- the base of the degree.
n- exponent.
If a n=1, which means the number a taken once and respectively: $a^n= 1$.
If a n=0, then $a^0= 1$.

Why this happens, we can find out when we get acquainted with the rules for multiplying and dividing powers.

multiplication rules

a) If powers with the same base are multiplied.
To $a^n * a^m$, we write the powers as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( a * a * \ldots * a )_(m )$.
The figure shows that the number a have taken n+m times, then $a^n * a^m = a^(n + m)$.

Example.
$2^3 * 2^2 = 2^5 = 32$.

This property is convenient to use to simplify the work when raising a number to a large power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.

b) If powers are multiplied with a different base, but the same exponent.
To $a^n * b^n$, we write the powers as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( b * b * \ldots * b )_(m )$.
If we swap the factors and count the resulting pairs, we get: $\underbrace( (a * b) * (a * b) * \ldots * (a * b) )_(n)$.

So $a^n * b^n= (a * b)^n$.

Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.

division rules

a) The base of the degree is the same, the exponents are different.
Consider dividing a degree with a larger exponent by dividing a degree with a smaller exponent.

So, it is necessary $\frac(a^n)(a^m)$, where n>m.

We write the degrees as a fraction:

$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( a * a * \ldots * a )_(m))$.

For convenience, we write the division as a simple fraction.

Now let's reduce the fraction.

It turns out: $\underbrace( a * a * \ldots * a )_(n-m)= a^(n-m)$.
Means, $\frac(a^n)(a^m)=a^(n-m)$.

This property will help explain the situation with raising a number to a power of zero. Let's assume that n=m, then $a^0= a^(n-n)=\frac(a^n)(a^n) =1$.

Examples.
$\frac(3^3)(3^2)=3^(3-2)=3^1=3$.

$\frac(2^2)(2^2)=2^(2-2)=2^0=1$.

b) The bases of the degree are different, the indicators are the same.
Let's say you need $\frac(a^n)( b^n)$. We write the powers of numbers as a fraction:

$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( b * b * \ldots * b )_(n))$.

Let's imagine for convenience.

Using the property of fractions, we divide a large fraction into a product of small ones, we get.
$\underbrace( \frac(a)(b) * \frac(a)(b) * \ldots * \frac(a)(b) )_(n)$.
Accordingly: $\frac(a^n)( b^n)=(\frac(a)(b))^n$.

Example.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.

First level

Degree and its properties. Comprehensive Guide (2019)

Why are degrees needed? Where do you need them? Why do you need to spend time studying them?

To learn everything about degrees, what they are for, how to use your knowledge in everyday life, read this article.

And, of course, knowing the degrees will bring you closer to successfully passing the OGE or the Unified State Examination and entering the university of your dreams.

Let's go... (Let's go!)

Important note! If instead of formulas you see gibberish, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac).

FIRST LEVEL

Exponentiation is the same mathematical operation as addition, subtraction, multiplication or division.

Now I will explain everything in human language using very simple examples. Pay attention. Examples are elementary, but explain important things.

Let's start with addition.

There is nothing to explain here. You already know everything: there are eight of us. Each has two bottles of cola. How much cola? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written in a different way: . Mathematicians are cunning and lazy people. They first notice some patterns, and then come up with a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of bottles of cola and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table. Of course, you can do everything slower, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, prettier one:

And what other tricky counting tricks did lazy mathematicians come up with? Correctly - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise this number to the fifth power. For example, . Mathematicians remember that two to the fifth power is. And they solve such problems in their mind - faster, easier and without errors.

To do this, you only need remember what is highlighted in color in the table of powers of numbers. Believe me, it will make your life much easier.

By the way, why is the second degree called square numbers, and the third cube? What does it mean? A very good question. Now you will have both squares and cubes.

Real life example #1

Let's start with a square or the second power of a number.

Imagine a square pool measuring meters by meters. The pool is in your backyard. It's hot and I really want to swim. But ... a pool without a bottom! It is necessary to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the area of ​​the bottom of the pool.

You can simply count by poking your finger that the bottom of the pool consists of cubes meter by meter. If your tiles are meter by meter, you will need pieces. It's easy... But where did you see such a tile? The tile will rather be cm by cm. And then you will be tormented by “counting with your finger”. Then you have to multiply. So, on one side of the bottom of the pool, we will fit tiles (pieces) and on the other, too, tiles. Multiplying by, you get tiles ().

Did you notice that we multiplied the same number by itself to determine the area of ​​the bottom of the pool? What does it mean? Since the same number is multiplied, we can use the exponentiation technique. (Of course, when you have only two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising to a power is much easier and there are also fewer errors in calculations. For the exam, this is very important).
So, thirty to the second degree will be (). Or you can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of some number. A square is an image of the second power of a number.

Real life example #2

Here's a task for you, count how many squares are on the chessboard using the square of the number ... On one side of the cells and on the other too. To count their number, you need to multiply eight by eight, or ... if you notice that a chessboard is a square with a side, then you can square eight. Get cells. () So?

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Unexpected, right?) Draw a pool: a bottom a meter in size and a meter deep and try to calculate how many meter by meter cubes will enter your pool.

Just point your finger and count! One, two, three, four…twenty-two, twenty-three… How much did it turn out? Didn't get lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes ... Easier, right?

Now imagine how lazy and cunning mathematicians are if they make that too easy. Reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself ... And what does this mean? This means that you can use the degree. So, what you once counted with a finger, they do in one action: three in a cube is equal. It is written like this:

Remains only memorize the table of degrees. Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can keep counting with your finger.

Well, in order to finally convince you that the degrees were invented by loafers and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Real life example #4

You have a million rubles. At the beginning of each year, you earn another million for every million. That is, each of your million at the beginning of each year doubles. How much money will you have in years? If you are now sitting and “counting with your finger”, then you are a very hardworking person and .. stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two times two ... in the second year - what happened, by two more, in the third year ... Stop! You noticed that the number is multiplied by itself once. So two to the fifth power is a million! Now imagine that you have a competition and the one who calculates faster will get these millions ... Is it worth remembering the degrees of numbers, what do you think?

Real life example #5

You have a million. At the beginning of each year, you earn two more for every million. It's great right? Every million is tripled. How much money will you have in a year? Let's count. The first year - multiply by, then the result by another ... It's already boring, because you already understood everything: three is multiplied by itself times. So the fourth power is a million. You just need to remember that three to the fourth power is or.

Now you know that by raising a number to a power, you will make your life much easier. Let's take a further look at what you can do with degrees and what you need to know about them.

Terms and concepts ... so as not to get confused

So, first, let's define the concepts. What do you think, what is exponent? It's very simple - this is the number that is "at the top" of the power of the number. Not scientific, but clear and easy to remember ...

Well, at the same time, what such a base of degree? Even simpler is the number that is at the bottom, at the base.

Here's a picture for you to be sure.

Well, in general terms, in order to generalize and remember better ... A degree with a base "" and an indicator "" is read as "in the degree" and is written as follows:

Power of a number with a natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is natural number? Elementary! Natural numbers are those that are used in counting when listing items: one, two, three ... When we count items, we don’t say: “minus five”, “minus six”, “minus seven”. We don't say "one third" or "zero point five tenths" either. These are not natural numbers. What do you think these numbers are?

Numbers like "minus five", "minus six", "minus seven" refer to whole numbers. In general, integers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and a number. Zero is easy to understand - this is when there is nothing. And what do negative ("minus") numbers mean? But they were invented primarily to denote debts: if you have a balance on your phone in rubles, this means that you owe the operator rubles.

All fractions are rational numbers. How did they come about, do you think? Very simple. Several thousand years ago, our ancestors discovered that they did not have enough natural numbers to measure length, weight, area, etc. And they came up with rational numbers… Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, then you get an irrational number.

Summary:

Let's define the concept of degree, the exponent of which is a natural number (that is, integer and positive).

1. Any number to the first power is equal to itself:
2. To square a number is to multiply it by itself:
3. To cube a number is to multiply it by itself three times:

Definition. To raise a number to a natural power is to multiply the number by itself times:
.

Degree properties

Where did these properties come from? I will show you now.

Let's see what is and ?

A-priory:

How many multipliers are there in total?

It's very simple: we added factors to the factors, and the result is factors.

But by definition, this is the degree of a number with an exponent, that is: , which was required to be proved.

Example: Simplify the expression.

Decision:

Example: Simplify the expression.

Decision: It is important to note that in our rule necessarily must be the same reason!
Therefore, we combine the degrees with the base, but remain a separate factor:

only for products of powers!

Under no circumstances should you write that.

2. that is -th power of a number

Just as with the previous property, let's turn to the definition of the degree:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:

Let's recall the formulas for abbreviated multiplication: how many times did we want to write?

But that's not true, really.

Degree with a negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the basis?

In degrees from natural indicator the basis may be any number. Indeed, we can multiply any number by each other, whether they are positive, negative, or even.

Let's think about what signs ("" or "") will have degrees of positive and negative numbers?

For example, will the number be positive or negative? BUT? ? With the first, everything is clear: no matter how many positive numbers we multiply with each other, the result will be positive.

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a minus gives a plus.” That is, or. But if we multiply by, it turns out.

Determine for yourself what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, I hope everything is clear? We simply look at the base and exponent, and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive.

Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple!

6 practice examples

Analysis of the solution 6 examples

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were swapped, the rule could apply.

But how to do that? It turns out that it is very easy: the even degree of the denominator helps us here.

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

whole we name the natural numbers, their opposites (that is, taken with the sign "") and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, we ask ourselves: why is this so?

Consider some power with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. What number must be multiplied by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

Let's go further. In addition to natural numbers and numbers, integers include negative numbers. To understand what a negative degree is, let's do the same as last time: we multiply some normal number by the same in a negative degree:

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

A number to a negative power is the inverse of the same number to a positive power. But at the same time base cannot be null:(because it is impossible to divide).

Let's summarize:

I. Expression is not defined in case. If, then.

II. Any number to the zero power is equal to one: .

III. A number that is not equal to zero to a negative power is the inverse of the same number to a positive power: .

Tasks for independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in the exam!

Let's continue to expand the range of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is "fractional degree" Let's consider a fraction:

Let's raise both sides of the equation to a power:

Now remember the rule "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal.

That is, the root of the th degree is the inverse operation of exponentiation: .

It turns out that. Obviously, this special case can be extended: .

Now add the numerator: what is it? The answer is easy to get with the power-to-power rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here a problem arises.

The number can be represented as other, reduced fractions, for example, or.

And it turns out that it exists, but does not exist, and these are just two different records of the same number.

Or another example: once, then you can write it down. But as soon as we write the indicator in a different way, we again get trouble: (that is, we got a completely different result!).

To avoid such paradoxes, consider only positive base exponent with fractional exponent.

So if:

• - natural number;
• is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

All the rules and properties of degrees here are exactly the same as for degrees with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero power- this is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number;

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number.

But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a degree to a degree:

Now look at the score. Does he remind you of anything? We recall the formula for abbreviated multiplication of the difference of squares:

AT this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal or both ordinary. We get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Definition of degree

The degree is an expression of the form: , where:

• — base of degree;
• - exponent.

Degree with natural exponent (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

• - natural number;
• is an integer;

Examples:

Degree properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression, the following product is obtained:

But by definition, this is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to note that in our rule necessarily must have the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

Another important note: this rule - only for products of powers!

Under no circumstances should I write that.

Just as with the previous property, let's turn to the definition of the degree:

Let's rearrange it like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the -th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be indicator degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

Indeed, we can multiply any number by each other, whether they are positive, negative, or even. Let's think about what signs ("" or "") will have degrees of positive and negative numbers?

For example, will the number be positive or negative? BUT? ?

With the first, everything is clear: no matter how many positive numbers we multiply with each other, the result will be positive.

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a minus gives a plus.” That is, or. But if we multiply by (), we get -.

And so on ad infinitum: with each subsequent multiplication, the sign will change. You can formulate these simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any power is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent, and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them into each other, divide them into pairs and get:

Before analyzing the last rule, let's solve a few examples.

Calculate the values ​​of expressions:

Solutions :

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares!

We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were reversed, rule 3 could be applied. But how to do this? It turns out that it is very easy: the even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it looks like this:

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced by by changing only one objectionable minus to us!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average level, we will analyze the degree with an irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with a negative integer - it's as if a certain “reverse process” has occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians have created to extend the concept of a degree to the entire space of numbers.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number. But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

1. Remember the difference of squares formula. Answer: .
2. We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
3. Nothing special, we apply the usual properties of degrees:

SECTION SUMMARY AND BASIC FORMULA

Degree is called an expression of the form: , where:

Degree with integer exponent

degree, the exponent of which is a natural number (i.e. integer and positive).

Degree with rational exponent

degree, the indicator of which is negative and fractional numbers.

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree properties

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any power is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!