The lesson will consider a more generalized version of the multiplication of fractions - this is exponentiation. First of all, we will talk about the natural degree of the fraction and examples that demonstrate similar actions with fractions. At the beginning of the lesson, we will also repeat the raising to a natural power of integer expressions and see how this is useful for solving further examples.

Topic: Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson: Raising an Algebraic Fraction to a Power

1. Rules for raising fractions and integer expressions to natural powers with elementary examples

The rule for raising ordinary and algebraic fractions to natural powers:

You can draw an analogy with the degree of an integer expression and remember what is meant by raising it to a power:

Example 1 .

As you can see from the example, raising a fraction to a power is a special case of multiplying fractions, which was studied in the previous lesson.

Example 2. a), b) - minus goes away, because we raised the expression to an even power.

For the convenience of working with degrees, we recall the basic rules for raising to a natural power:

- product of degrees;

- division of degrees;

Raising a degree to a power;

The degree of the work.

Example 3. - this is known to us since the topic "Raising to the power of integer expressions", except for one case: it does not exist.

2. The simplest examples for raising algebraic fractions to natural powers

Example 4. Raise a fraction to a power.

Solution. When raised to an even power, minus goes away:

Example 5. Raise a fraction to a power.

Solution. Now we use the rules for raising a degree to a power immediately without a separate schedule:

.

Now consider the combined tasks in which we will need to raise fractions to a power, and multiply them, and divide.

Example 6: Perform actions.

Solution. . Next, you need to make a reduction. We will describe once in detail how we will do this, and then we will indicate the result immediately by analogy:. Similarly (or according to the rule of division of degrees). We have: .

Example 7: Perform actions.

Solution. . The reduction is carried out by analogy with the example discussed earlier.

Example 8: Perform actions.

Solution. . In this example, we once again described the process of reducing powers in fractions in more detail in order to consolidate this method.

3. More complex examples for raising algebraic fractions to natural powers (taking into account signs and with terms in brackets)

Example 9: Perform actions .

Solution. In this example, we will already skip the separate multiplication of fractions, and immediately use the rule for their multiplication and write it down under one denominator. At the same time, we follow the signs - in this case, the fractions are raised to even powers, so the minuses disappear. Let's do a reduction at the end.

Example 10: Perform actions .

Solution. In this example, there is a division of fractions, remember that in this case the first fraction is multiplied by the second, but inverted.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * ... * an = an.

For example, a=2, n=3: 2 * 2 * 2=2^3 = 8 .

In general, exponentiation is often used in various formulas in mathematics and physics. This function has a more scientific purpose than the four basic ones: Addition, Subtraction, Multiplication, Division.

Raising a number to a power

Raising a number to a power is not a difficult operation. It is related to multiplication like the relationship between multiplication and addition. Record an - a short record of the n-th number of numbers "a" multiplied by each other.

Consider exponentiation on the simplest examples, moving on to complex ones.

For example, 42. 42 = 4 * 4 = 16 . Four squared (to the second power) equals sixteen. If you do not understand the multiplication 4 * 4, then read our article about multiplication.

Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 . Five cubed (to the third power) equals one hundred and twenty-five.

Another example: 9^3. 9^3 = 9 * 9 * 9 = 81 * 9 = 729 . Nine cubed equals seven hundred twenty-nine.

Exponentiation Formulas

To correctly raise to a power, you need to remember and know the formulas below. There is nothing beyond natural in this, the main thing is to understand the essence and then they will not only be remembered, but also seem easy.

Raising a monomial to a power

What is a monomial? This is the product of numbers and variables in any quantity. For example, two is a monomial. And this article is about raising such monomials to a power.

Using exponentiation formulas, it will not be difficult to calculate the exponentiation of a monomial to a power.

For example, (3x^2y^3)^2= 3^2 * x^2 * 2 * y^(3 * 2) = 9x^4y^6; If you raise a monomial to a power, then each component of the monomial is raised to a power.

When raising a variable that already has a degree to a power, the degrees are multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6 ;

Raising to a negative power

A negative exponent is the reciprocal of a number. What is a reciprocal? For any number X, the reciprocal is 1/X. That is X-1=1/X. This is the essence of the negative degree.

Consider the example (3Y)^-3:

(3Y)^-3 = 1/(27Y^3).

Why is that? Since there is a minus in the degree, we simply transfer this expression to the denominator, and then raise it to the third power. Just right?

Raising to a fractional power

Let's start with a specific example. 43/2. What does power 3/2 mean? 3 - numerator, means raising a number (in this case 4) to a cube. The number 2 is the denominator, this is the extraction of the second root of the number (in this case 4).

Then we get the square root of 43 = 2^3 = 8 . Answer: 8.

So, the denominator of a fractional degree can be either 3 or 4, and to infinity any number, and this number determines the degree of the square root extracted from a given number. Of course, the denominator cannot be zero.

Raising a root to a power

If the root is raised to a power equal to the power of the root itself, then the answer is the radical expression. For example, (√x)2 = x. And so in any case of equality of the degree of the root and the degree of raising the root.

If (√x)^4. Then (√x)^4=x^2. To check the solution, we translate the expression into an expression with a fractional degree. Since the root is square, the denominator is 2. And if the root is raised to the fourth power, then the numerator is 4. We get 4/2=2. Answer: x = 2.

In any case, the best option is to simply convert the expression to a fractional exponent. If the fraction is not reduced, then such an answer will be, provided that the root of the given number is not allocated.

Exponentiation of a complex number

What is a complex number? A complex number is an expression that has the formula a + b * i; a, b are real numbers. i is the number that, when squared, gives the number -1.

Consider an example. (2 + 3i)^2.

(2 + 3i)^2 = 22 +2 * 2 * 3i +(3i)^2 = 4+12i^-9=-5+12i.

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Exponentiation Grade 7

Raising to a power begins to pass schoolchildren only in the seventh grade.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * … * an=an .

For example, a=2, n=3: 2 * 2 * 2 = 2^3 = 8.

Solution Examples:

Exponentiation presentation

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In continuation of the conversation about the degree of a number, it is logical to deal with finding the value of the degree. This process has been named exponentiation. In this article, we will just study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And by tradition, we will consider in detail the solutions to examples of raising numbers to various degrees.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation is to find the value of the power of a number.

Thus, finding the value of the power of a with the exponent r and raising the number a to the power of r is the same thing. For example, if the task is “calculate the value of the power (0.5) 5”, then it can be reformulated as follows: “Raise the number 0.5 to the power of 5”.

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising the number a to a fractional power m / n, the root of the nth degree from the number a is first extracted, after which the result is raised to an integer power m.

Consider solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We show two solutions.

First way. By definition of degree with a fractional exponent. We calculate the value of the degree under the sign of the root, after which we extract the cube root: .

The second way. By definition of a degree with a fractional exponent and on the basis of the properties of the roots, the equalities are true . Now extract the root Finally, we raise to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that the fractional exponent can be written as a decimal fraction or a mixed number, in these cases it should be replaced by the corresponding ordinary fraction, and then exponentiation should be performed.

Example.

Calculate (44.89) 2.5 .

Solution.

We write the exponent in the form of an ordinary fraction (if necessary, see the article): . Now we perform raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather laborious process (especially when the numerator and denominator of the fractional exponent are quite large numbers), which is usually carried out using computer technology.

In conclusion of this paragraph, we will dwell on the construction of the number zero to a fractional power. We gave the following meaning to the fractional degree of zero of the form: for we have , while zero to the power m/n is not defined. So, zero to a positive fractional power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions and 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the degree of a number with an irrational exponent. In this case, for practical purposes, it is usually sufficient to obtain the value of the degree up to a certain sign. We note right away that in practice this value is calculated using electronic computing technology, since manual raising to an irrational power requires a large number of cumbersome calculations. But nevertheless we will describe in general terms the essence of the actions.

To get an approximate value of the exponent of a with an irrational exponent, some decimal approximation of the exponent is taken, and the value of the exponent is calculated. This value is the approximate value of the degree of the number a with an irrational exponent. The more accurate the decimal approximation of the number is taken initially, the more accurate the degree value will be in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of an irrational indicator: . Now we raise 2 to a rational power of 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈ 2.250116. In this way, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of an irrational exponent, for example, , then we get a more accurate value of the original degree: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

We figured out what the degree of a number is in general. Now we need to understand how to correctly calculate it, i.e. raise numbers to powers. In this material, we will analyze the basic rules for calculating the degree in the case of an integer, natural, fractional, rational and irrational exponent. All definitions will be illustrated with examples.

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The concept of exponentiation

Let's start with the formulation of basic definitions.

Definition 1

Exponentiation is the calculation of the value of the power of some number.

That is, the words "calculation of the value of the degree" and "exponentiation" mean the same thing. So, if the task is "Raise the number 0 , 5 to the fifth power", this should be understood as "calculate the value of the power (0 , 5) 5 .

Now we give the basic rules that must be followed in such calculations.

Recall what a power of a number with a natural exponent is. For a power with base a and exponent n, this will be the product of the nth number of factors, each of which is equal to a. This can be written like this:

To calculate the value of the degree, you need to perform the operation of multiplication, that is, multiply the bases of the degree the specified number of times. The very concept of a degree with a natural indicator is based on the ability to quickly multiply. Let's give examples.

Example 1

Condition: Raise - 2 to the power of 4 .

Solution

Using the definition above, we write: (− 2) 4 = (− 2) (− 2) (− 2) (− 2) . Next, we just need to follow these steps and get 16 .

Let's take a more complicated example.

Example 2

Calculate the value 3 2 7 2

Solution

This entry can be rewritten as 3 2 7 · 3 2 7 . Earlier we looked at how to correctly multiply the mixed numbers mentioned in the condition.

Perform these steps and get the answer: 3 2 7 3 2 7 = 23 7 23 7 = 529 49 = 10 39 49

If the task indicates the need to raise irrational numbers to a natural power, we will need to first round their bases to a digit that will allow us to get an answer of the desired accuracy. Let's take an example.

Example 3

Perform the squaring of the number π .

Solution

Let's round it up to hundredths first. Then π 2 ≈ (3, 14) 2 = 9, 8596. If π ≈ 3 . 14159, then we will get a more accurate result: π 2 ≈ (3, 14159) 2 = 9, 8695877281.

Note that the need to calculate the powers of irrational numbers in practice arises relatively rarely. We can then write the answer as the power itself (ln 6) 3 or convert if possible: 5 7 = 125 5 .

Separately, it should be indicated what the first power of a number is. Here you can just remember that any number raised to the first power will remain itself:

This is clear from the record. .

It does not depend on the basis of the degree.

Example 4

So, (− 9) 1 = − 9 , and 7 3 raised to the first power remains equal to 7 3 .

For convenience, we will analyze three cases separately: if the exponent is a positive integer, if it is zero, and if it is a negative integer.

In the first case, this is the same as raising to a natural power: after all, positive integers belong to the set of natural numbers. We have already described how to work with such degrees above.

Now let's see how to properly raise to the zero power. With a base that is non-zero, this calculation always produces an output of 1 . We have previously explained that the 0th power of a can be defined for any real number not equal to 0 , and a 0 = 1 .

Example 5

5 0 = 1 , (- 2 , 56) 0 = 1 2 3 0 = 1

0 0 - not defined.

We are left with only the case of a degree with a negative integer exponent. We have already discussed that such degrees can be written as a fraction 1 a z, where a is any number, and z is a negative integer. We see that the denominator of this fraction is nothing more than an ordinary degree with a positive integer, and we have already learned how to calculate it. Let's give examples of tasks.

Example 6

Raise 3 to the -2 power.

Solution

Using the definition above, we write: 2 - 3 = 1 2 3

We calculate the denominator of this fraction and get 8: 2 3 \u003d 2 2 2 \u003d 8.

Then the answer is: 2 - 3 = 1 2 3 = 1 8

Example 7

Raise 1, 43 to the -2 power.

Solution

Reformulate: 1 , 43 - 2 = 1 (1 , 43) 2

We calculate the square in the denominator: 1.43 1.43. Decimals can be multiplied in this way:

As a result, we got (1, 43) - 2 = 1 (1, 43) 2 = 1 2 , 0449 . It remains for us to write this result in the form of an ordinary fraction, for which it is necessary to multiply it by 10 thousand (see the material on the conversion of fractions).

Answer: (1, 43) - 2 = 10000 20449

A separate case is raising a number to the minus first power. The value of such a degree is equal to the number opposite to the original value of the base: a - 1 \u003d 1 a 1 \u003d 1 a.

Example 8

Example: 3 − 1 = 1 / 3

9 13 - 1 = 13 9 6 4 - 1 = 1 6 4 .

How to raise a number to a fractional power

To perform such an operation, we need to recall the basic definition of a degree with a fractional exponent: a m n \u003d a m n for any positive a, integer m and natural n.

Definition 2

Thus, the calculation of a fractional degree must be performed in two steps: raising to an integer power and finding the root of the nth degree.

We have the equality a m n = a m n , which, given the properties of the roots, is usually used to solve problems in the form a m n = a n m . This means that if we raise the number a to a fractional power m / n, then first we extract the root of the nth degree from a, then we raise the result to a power with an integer exponent m.

Let's illustrate with an example.

Example 9

Calculate 8 - 2 3 .

Solution

Method 1. According to the basic definition, we can represent this as: 8 - 2 3 \u003d 8 - 2 3

Now let's calculate the degree under the root and extract the third root from the result: 8 - 2 3 = 1 64 3 = 1 3 3 64 3 = 1 3 3 4 3 3 = 1 4

Method 2. Let's transform the basic equality: 8 - 2 3 \u003d 8 - 2 3 \u003d 8 3 - 2

After that, we extract the root 8 3 - 2 = 2 3 3 - 2 = 2 - 2 and square the result: 2 - 2 = 1 2 2 = 1 4

We see that the solutions are identical. You can use any way you like.

There are cases when the degree has an indicator expressed as a mixed number or decimal fraction. For ease of calculation, it is better to replace it with an ordinary fraction and count as indicated above.

Example 10

Raise 44.89 to the power of 2.5.

Solution

Let's convert the value of the indicator into an ordinary fraction - 44, 89 2, 5 = 49, 89 5 2.

And now we perform all the actions indicated above in order: 44 , 89 5 2 = 44 , 89 5 = 44 , 89 5 = 4489 100 5 = 4489 100 5 = 67 2 10 2 5 = 67 10 5 = = 1350125107 100000 = 13 501, 25107

Answer: 13501, 25107.

If there are large numbers in the numerator and denominator of a fractional exponent, then calculating such exponents with rational exponents is a rather difficult job. It usually requires computer technology.

Separately, we dwell on the degree with a zero base and a fractional exponent. An expression of the form 0 m n can be given the following meaning: if m n > 0, then 0 m n = 0 m n = 0 ; if m n< 0 нуль остается не определен. Таким образом, возведение нуля в дробную положительную степень приводит к нулю: 0 7 12 = 0 , 0 3 2 5 = 0 , 0 0 , 024 = 0 , а в целую отрицательную - значения не имеет: 0 - 4 3 .

How to raise a number to an irrational power

The need to calculate the value of the degree, in the indicator of which there is an irrational number, does not arise so often. In practice, the task is usually limited to calculating an approximate value (up to a certain number of decimal places). This is usually calculated on a computer due to the complexity of such calculations, so we will not dwell on this in detail, we will only indicate the main provisions.

If we need to calculate the value of the degree a with an irrational exponent a , then we take the decimal approximation of the exponent and count from it. The result will be an approximate answer. The more accurate the decimal approximation taken, the more accurate the answer. Let's show with an example:

Example 11

Compute an approximate value of 21 , 174367 ....

Solution

We restrict ourselves to the decimal approximation a n = 1 , 17 . Let's do the calculations using this number: 2 1 , 17 ≈ 2 , 250116 . If we take, for example, the approximation a n = 1 , 1743 , then the answer will be a little more precise: 2 1 , 174367 . . . ≈ 2 1 . 1743 ≈ 2 . 256833 .

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