Raising a fraction to a cube. Raising an algebraic fraction to a power

It's time to familiarize yourself with erection algebraic fraction to a degree. This action with algebraic fractions, in terms of the degree, is reduced to multiplication identical fractions. In this article, we will give the corresponding rule, and consider examples of raising algebraic fractions to natural powers.

Page navigation.

The rule of raising an algebraic fraction to a power, its proof

Before talking about raising an algebraic fraction to a power, it does not hurt to remember what the product of the same factors that stand at the base of the degree is, and their number is determined by the indicator. For example, 2 3 =2 2 2=8 .

And now let's remember the rule of raising to the power of an ordinary fraction - for this you need to separately raise the numerator to the indicated power, and separately the denominator. For instance, . This rule applies to raising an algebraic fraction to a natural power.

Raising an algebraic fraction to a natural power gives a new fraction, in the numerator of which is the specified degree of the numerator of the original fraction, and in the denominator - the degree of the denominator. In literal form, this rule corresponds to the equality , where a and b are arbitrary polynomials (in particular cases, monomials or numbers), and b is a nonzero polynomial, and n is .

The proof of the voiced rule for raising an algebraic fraction to a power is based on the definition of a degree with a natural exponent and on how we defined the multiplication of algebraic fractions: .

Examples, Solutions

The rule obtained in the previous paragraph reduces the raising of an algebraic fraction to a power to the raising of the numerator and denominator of the original fraction to this power. And since the numerator and denominator of the original algebraic fraction are polynomials (in the particular case, monomials or numbers), the original task is reduced to raising polynomials to a power. After performing this action, a new algebraic fraction will be obtained, identically equal to the specified power of the original algebraic fraction.

Let's take a look at a few examples.

Example.

Square an algebraic fraction.

Solution.

Let's write the degree. Now we turn to the rule for raising an algebraic fraction to a power, it gives us the equality . It remains to convert the resulting fraction to the form of an algebraic fraction by raising monomials to a power. So .

Usually, when raising an algebraic fraction to a power, the course of the solution is not explained, and the solution is written briefly. Our example corresponds to the record .

Answer:

When polynomials, especially binomials, are in the numerator and / or denominator of an algebraic fraction, then when raising it to a power, it is advisable to use the corresponding abbreviated multiplication formulas.

Example.

Raise an algebraic fraction to the second degree.

Solution.

By the rule of raising a fraction to a power, we have .

To transform the resulting expression in the numerator, we use difference squared formula, and in the denominator - the formula of the square of the sum of three terms:

Answer:

In conclusion, we note that if we raise an irreducible algebraic fraction to a natural power, then the result will also be an irreducible fraction. If the original fraction is reducible, then before raising it to a power, it is advisable to reduce the algebraic fraction so as not to perform the reduction after raising to a power.

Bibliography.

Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Mordkovich A. G. Algebra. 8th grade. At 2 p.m. Part 1. Student's textbook educational institutions/ A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

All rights reserved.
Protected by copyright law. No part of www.website, including internal materials And external design may not be reproduced in any form or used without the prior written permission of the copyright holder.

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 touching 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 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 contain enough big 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 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 this value is calculated in practice using electronic computing technology, since raising to ir rational degree manually requires a large number cumbersome calculations. However, we will describe in general terms essence of action.

To get an approximate value of the power of a with irrational indicator, 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 decimal approximation of a number is taken initially, the more exact value degree will be obtained 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).

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 special case multiplication of 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 in more detail the process of reducing powers in fractions 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.

The topic boils down to the fact that we need to multiply identical fractions. This article will tell you what rule you need to use in order to correctly raise algebraic fractions to natural powers.

Yandex.RTB R-A-339285-1

The rule for raising an algebraic fraction to a power, its proof

Before you start raising to a power, you need to deepen your knowledge with the help of an article about a degree with a natural indicator, where there is a product of identical factors that are at the base of the degree, and their number is determined by the indicator. For example, the number 2 3 \u003d 2 2 2 \u003d 8.

When raising to a power, we most often use the rule. To do this, separately raise the numerator and the denominator separately. Consider the example 2 3 2 = 2 2 3 2 = 4 9 . The rule applies to raising a fraction to a natural power.

At raising an algebraic fraction to a natural power we get a new one, where the numerator has the degree of the original fraction, and the denominator has the degree of the denominator. This is all of the form a b n = a n b n , where a and b are arbitrary polynomials, b is non-zero, and n is a natural number.

The proof of this rule is written as a fraction, which must be raised to a power, based on the definition itself with a natural indicator. Then we get the multiplication of fractions of the form a b n = a b · a b · . . . · a b = a · a · . . . · a b · b · . . . b = a n b n

Examples, Solutions

The rule for raising an algebraic fraction to a power is performed sequentially: first the numerator, then the denominator. When there is a polynomial in the numerator and denominator, then the task itself will be reduced to raising the given polynomial to a power. After that, a new fraction will be indicated, which is equal to the original one.

Example 1

Squaring the fraction x 2 3 y z 3

Solution

It is necessary to fix the degree x 2 3 · y · z 3 2 . According to the rule of raising an algebraic fraction to a power, we obtain an equality of the form x 2 3 · y · z 3 2 = x 2 2 3 · y · z 3 2 . Now it is necessary to convert the resulting fraction to an algebraic form by exponentiation. Then we get an expression of the form

x 2 2 3 y z 3 2 = x 2 2 3 2 y 2 z 3 2 = x 4 9 y 2 z 6

All cases of exponentiation do not require a detailed explanation, so the solution itself has a short record. That is, we get that

x 2 3 y z 3 2 = x 2 2 3 y z 3 2 = x 4 9 y 2 z 6

Answer: x 2 3 y z 3 2 = x 4 9 y 2 z 6 .

If the numerator and denominator have polynomials, then it is necessary to raise the whole fraction to a power, and then apply the abbreviated multiplication formulas to simplify it.

Example 2

Square the fraction 2 x - 1 x 2 + 3 x y - y.

Solution

From the rule we have that

2 x - 1 x 2 + 3 x y - y 2 = 2 x - 1 2 x 2 + 3 x y - y 2

To convert the expression, you must use the formula for the square of the sum of three terms in the denominator, and in the numerator - the square of the difference, which will simplify the expression. We get:

2 x - 1 2 x 2 + 3 x y - y 2 = = 2 x 2 - 2 2 x 1 + 1 2 x 2 2 + 3 x y 2 + - y 2 + 2 x 2 3 x y + 2 x 2 (- y) + 2 3 x y - y = = 4 x 2 - 4 x + 1 x 4 + 9 x 2 y 2 + y 2 + 6 x 3 y - 2 x 2 y - 6 x y 2

Answer: 2 x - 1 2 x 2 + 3 x y - y 2 = 4 x 2 - 4 x + 1 x 4 + 9 x 2 y 2 + y 2 + 6 x 3 y - 2 x 2 y - 6 x y 2

Note that when raising a fraction that we cannot reduce to a natural power, we also obtain an irreducible fraction. This does not make it easier to solve further. When a given fraction can be reduced, then when exponentiated, we find that it is necessary to perform the reduction of the algebraic fraction, in order to avoid performing the reduction after raising to the power.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

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.

Yandex.RTB R-A-339285-1

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 but 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

Convert the value of the indicator to common 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.

Let us dwell separately 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 accurate: 2 1 , 174367 . . . ≈ 2 1 . 1743 ≈ 2 . 256833 .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter