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. Due to the bulkiness, it was thought to reduce the notation to 3 * 100 = 300. In fact, the notation “three times one hundred” means that you need to take one hundred triples 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 sage 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 bases is equal to this base to a degree equal to the quotient of the dividend and divisor: a ^ n: a ^ m = 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.
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.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.
In the last video tutorial, we learned that the degree of a certain base is an expression that is the product of the base and itself, taken in an amount equal to the exponent. Let us now study some of the most important properties and operations of powers.
For example, let's multiply two different powers with the same base:
Let's take a look at this piece in its entirety:
(2) 3 * (2) 2 = (2)*(2)*(2)*(2)*(2) = 32
Calculating the value of this expression, we get the number 32. On the other hand, as can be seen from the same example, 32 can be represented as a product of the same base (two), taken 5 times. And indeed, if you count, then:
Thus, it can be safely concluded that:
(2) 3 * (2) 2 = (2) 5
This rule works successfully for any indicators and any grounds. This property of multiplication of the degree follows from the rule of preservation of the meaning of expressions during transformations in the product. For any base a, the product of two expressions (a) x and (a) y is equal to a (x + y). In other words, when producing any expressions with the same base, the final monomial has a total degree formed by adding the degree of the first and second expressions.
The presented rule also works great when multiplying several expressions. The main condition is that the bases for all be the same. For example:
(2) 1 * (2) 3 * (2) 4 = (2) 8
It is impossible to add degrees, and in general to carry out any power joint actions with two elements of the expression, if their bases are different.
As our video shows, due to the similarity of the processes of multiplication and division, the rules for adding powers during a product are perfectly transferred to the division procedure. Consider this example:
Let's make a term-by-term transformation of the expression into a full form and reduce the same elements in the dividend and divisor:
(2)*(2)*(2)*(2)*(2)*(2) / (2)*(2)*(2)*(2) = (2)(2) = (2) 2 = 4
The end result of this example is not so interesting, because already in the course of its solution it is clear that the value of the expression is equal to the square of two. And it is the deuce that is obtained by subtracting the degree of the second expression from the degree of the first.
To determine the degree of the quotient, it is necessary to subtract the degree of the divisor from the degree of the dividend. The rule works with the same basis for all its values and for all natural powers. In abstract form, we have:
(a) x / (a) y = (a) x - y
The definition for the zero degree follows from the rule for dividing identical bases with powers. Obviously, the following expression is:
(a) x / (a) x \u003d (a) (x - x) \u003d (a) 0
On the other hand, if we divide in a more visual way, we get:
(a) 2 / (a) 2 = (a) (a) / (a) (a) = 1
When reducing all visible elements of a fraction, the expression 1/1 is always obtained, that is, one. Therefore, it is generally accepted that any base raised to the zero power is equal to one:
Regardless of the value of a.
However, it would be absurd if 0 (which still gives 0 for any multiplication) is somehow equal to one, so an expression like (0) 0 (zero to the zero degree) simply does not make sense, and to formula (a) 0 = 1 add a condition: "if a is not equal to 0".
Let's do the exercise. Let's find the value of the expression:
(34) 7 * (34) 4 / (34) 11
Since the base is the same everywhere and equals 34, the final value will have the same base with a degree (according to the above rules):
In other words:
(34) 7 * (34) 4 / (34) 11 = (34) 0 = 1
Answer: The expression is equal to one.
The concept of a degree in mathematics is introduced as early as the 7th grade in an algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values and the ability to correctly and quickly count. For faster and better work with mathematics degrees, they came up with the properties of a degree. They help to cut down on big calculations, to convert a huge example into a single number to some degree. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.
degree properties
We will consider 12 properties of a degree, including properties of powers with the same base, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.
1st property.
Many people very often forget about this property, make mistakes, representing a number to the zero degree as zero.
2nd property.
3rd property.
It must be remembered that this property can only be used when multiplying numbers, it does not work with the sum! And we must not forget that this and the following properties apply only to powers with the same base.
4th property.
If the number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in brackets to correctly replace the sign in further calculations.
The property only works when dividing, not when subtracting!
5th property.
6th property.
This property can also be applied in reverse. A unit divided by a number to some degree is that number to a negative power.
7th property.
This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not the properties of the power.
8th property.
9th property.
This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will change depending on the denominator of the degree.
Also, this property is often used in reverse order. The root of any power of a number can be represented as that number to the power of one divided by the power of the root. This property is very useful in cases where the root of the number is not extracted.
10th property.
This property works not only with the square root and the second degree. If the degree of the root and the degree to which this root is raised are the same, then the answer will be a radical expression.
11th property.
You need to be able to see this property in time when solving it in order to save yourself from huge calculations.
12th property.
Each of these properties will meet you more than once in tasks, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of mathematical knowledge.
Application of degrees and their properties
They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as powers often complicate equations and examples related to other sections of mathematics. Exponents help to avoid large and long calculations, it is easier to reduce and calculate the exponents. But to work with large powers, or with powers of large numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time in solving by eliminating the need for long calculations.
The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.
Abbreviated multiplication formulas are another example of the use of powers. They cannot use the properties of degrees, they are decomposed according to special rules, but in each abbreviated multiplication formula there are invariably degrees.
Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and in the future, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations for conversions of units of measurement or calculations of problems, just as in physics, occur using the properties of the degree.
Degrees are also very useful in astronomy, where you can rarely find the use of the properties of a degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.
Degrees are also used in everyday life, when calculating areas, volumes, distances.
With the help of degrees, very large and very small values \u200b\u200bare written in any field of science.
exponential equations and inequalities
Degree properties occupy a special place precisely in exponential equations and inequalities. These tasks are very common, both in the school course and in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, therefore, knowing all the properties, it will not be difficult to solve such an equation or inequality.
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:
www.algebraclass.ru
Addition, subtraction, multiplication, and division of powers
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;
That's why, 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.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
(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.
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
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.
Please note that property No. 4, like other properties of degrees, is also applied in reverse order.
(a n b n)= (a b) n
That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged.
2 4 5 4 = (2 5) 4 = 10 4 = 10,000
0.5 16 2 16 = (0.5 2) 16 = 1
In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different exponents. In this case, we advise you to do the following.
For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216
Example of exponentiation of a decimal fraction.
4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = four
Properties 5
Power of the quotient (fractions)
To raise a quotient to a power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.
(a: b) n \u003d a n: b n, where "a", "b" are any rational numbers, b ≠ 0, n is any natural number.
(5: 3) 12 = 5 12: 3 12
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.
Degrees and Roots
Operations with powers and roots. Degree with negative ,
zero and fractional indicator. About expressions that don't make sense.
Operations with degrees.
1. When multiplying powers with the same base, their indicators are added up:
a m · a n = a m + n .
2. When dividing degrees with the same base, their indicators subtracted .
3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.
4. The degree of the ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):
(a/b) n = a n / b n .
5. When raising a degree to a power, their indicators are multiplied:
All of the above formulas are read and executed in both directions from left to right and vice versa.
EXAMPLE (2 3 5 / 15)² = 2 ² 3 ² 5 ² / 15 ² = 900 / 225 = 4 .
Operations with roots. In all the formulas below, the symbol means arithmetic root(radical expression is positive).
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of the ratio is equal to the ratio of the roots of the dividend and divisor:
3. When raising a root to a power, it is enough to raise to this power root number:
4. If you increase the degree of the root by m times and simultaneously raise the root number to the m -th degree, then the value of the root will not change:
5. If you reduce the degree of the root by m times and at the same time extract the root of the m-th degree from the radical number, then the value of the root will not change:
Extension of the concept of degree. So far, we have considered degrees only with a natural indicator; but operations with powers and roots can also lead to negative, zero and fractional indicators. All these exponents require an additional definition.
Degree with a negative exponent. The power of some number with a negative (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the negative exponent:
Now the formula a m : a n = a m-n can be used not only for m, more than n, but also at m, less than n .
EXAMPLE a 4: a 7 = a 4 — 7 = a — 3 .
If we want the formula a m : a n = a m — n was fair at m = n, we need a definition of the zero degree.
Degree with zero exponent. The degree of any non-zero number with zero exponent is 1.
EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.
A degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the root of the nth degree from the mth power of this number a:
About expressions that don't make sense. There are several such expressions.
where a ≠ 0 , does not exist.
Indeed, if we assume that x is a certain number, then, in accordance with the definition of the division operation, we have: a = 0· x, i.e. a= 0, which contradicts the condition: a ≠ 0
— any number.
Indeed, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 = 0 x. But this equality holds for any number x, which was to be proved.
0 0 — any number.
Solution. Consider three main cases:
1) x = 0 – this value does not satisfy this equation
2) when x> 0 we get: x / x= 1, i.e. 1 = 1, whence follows,
what x- any number; but taking into account that
our case x> 0 , the answer is x > 0 ;
Rules for multiplying powers with different bases
DEGREE WITH A RATIONAL INDICATOR,
POWER FUNCTION IV
§ 69. Multiplication and division of powers with the same bases
Theorem 1. To multiply powers with the same bases, it is enough to add the exponents, and leave the base the same, that is
Proof. By definition of degree
2 2 2 3 = 2 5 = 32; (-3) (-3) 3 = (-3) 4 = 81.
We have considered the product of two powers. In fact, the proved property is true for any number of powers with the same bases.
Theorem 2. To divide powers with the same bases, when the indicator of the dividend is greater than the indicator of the divisor, it is enough to subtract the indicator of the divisor from the indicator of the dividend, and leave the base the same, that is at t > n
(a =/= 0)
Proof. Recall that the quotient of dividing one number by another is the number that, when multiplied by a divisor, gives the dividend. Therefore, prove the formula , where a =/= 0, it's like proving the formula
If a t > n , then the number t - p will be natural; therefore, by Theorem 1
Theorem 2 is proved.
Note that the formula
proved by us only under the assumption that t > n . Therefore, from what has been proved, it is not yet possible to draw, for example, the following conclusions:
In addition, we have not yet considered degrees with negative exponents, and we do not yet know what meaning can be given to the expression 3 - 2 .
Theorem 3. To raise a power to a power, it is enough to multiply the exponents, leaving the base of the exponent the same, that is
Proof. Using the definition of degree and Theorem 1 of this section, we get:
Q.E.D.
For example, (2 3) 2 = 2 6 = 64;
518 (Oral.) Determine X from the equations:
1) 2 2 2 2 3 2 4 2 5 2 6 = 2 x ; 3) 4 2 4 4 4 6 4 8 4 10 = 2 x ;
2) 3 3 3 3 5 3 7 3 9 = 3 x ; 4) 1 / 5 1 / 25 1 / 125 1 / 625 = 1 / 5 x .
519. (Adjusted) Simplify:
520. (Adjusted) Simplify:
521. Present these expressions as degrees with the same bases:
1) 32 and 64; 3) 85 and 163; 5) 4 100 and 32 50;
2) -1000 and 100; 4) -27 and -243; 6) 81 75 8 200 and 3 600 4 150.
