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 degree.

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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 .

Now let's remember the exponentiation rule common fraction- for this you need to separately raise the numerator to the indicated power, and separately - the denominator. For example, . 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 raising 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.

Decision.

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.

Decision.

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 cancellable, 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.

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A fraction is the ratio of the numerator to the denominator, and the denominator must not be zero, and the numerator can be any.

When raising any fraction to an arbitrary power, you need to separately raise the numerator and denominator of the fraction to this power, after which we must count these powers and thus get the fraction raised to the power.

For example:

(2/7)^2 = 2^2/7^2 = 4/49

(2 / 3)^3 = (2 / 3) (2 / 3) (2 / 3) = 2^3 / 3^3

negative power

If we are dealing with a negative degree, then we must first “Reverse the fraction”, and only then raise it to a power according to the rule written above.

(2/7)^(-2) = (7/2)^2 = 7^2/2^2

Letter degree

When working with literal values ​​such as "x" and "y", exponentiation follows the same rule as before.

We can also check ourselves by raising the fraction ½ to the 3rd power, as a result we get ½ * ½ * ½ = 1/8 which is essentially the same as

(1/2)^3 = 1/8.

Literal exponentiation x^y

Multiplication and division of fractions with powers

If we multiply powers with the same base, then the base itself remains the same, and we add the exponents. If we divide powers with the same base, then the base of the degree also remains the same, and the exponents are subtracted.

This can be shown very easily with an example:

(3^23)*(3^8)=3^(23+8) = 3^31

(2^4)/(2^3) = 2^(4-3) = 2^1 = 2

We could get the same thing if we simply raised the denominator and numerator separately to the power of 3 and 4, respectively.

Raising a fraction with a power to another power

When raising a fraction, which is already in a power, once again into a power, we must first do the internal exponentiation and then move on to the external part of the exponentiation. In other words, we can simply simply multiply these powers and raise the fraction to the resulting power.

For example:

(2^4)^2 = 2^ 4 2 = 2^8

Uniting, square root

Also, we must not forget that raising absolutely any fraction to the zero power will give us 1, just like any other number, when raised to a power equal to zero, we will get 1.

Normal Square root can also be expressed as a fraction

Square root 3 = 3^(1/2)

If we are dealing with a square root under which there is a fraction, then we can represent this fraction in the numerator of which there will be a square root of 2 - degree (because the square root)

And the denominator will also contain the square root, i.e. in other words, we will see the ratio of two roots, this may be useful for solving some problems and examples.

If we raise a fraction that is under the square root to the second power, then we get the same fraction.

The product of two fractions under the same degree will be equal to the product of these two fractions, each of which individually will be under its own degree.

Remember: you can't divide by zero!

Also, do not forget about a very important remark for a fraction such as the denominator should not be equal to zero. In the future, in many equations, we will use this restriction, called ODZ - the range of acceptable values

When comparing two fractions with the same base, but varying degrees, the greater will be the fraction in which the degree will be greater, and the smaller one in which the degree will be less, if not only the bases are equal, but also the degrees, the fraction is considered the same.

Examples:

ex: 14^3.8 / 14^(-0.2) = 14^(3.8 -0.2) = 139.6

6^(1.77) 6^(- 0.75) = 6^(1.77+(- 0.75)) = 79.7 - 1.3 = 78.6

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 over 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.

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

Example 5. Raise a fraction to a power.

Decision. 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.

Decision. . 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.

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

Example 8: Perform actions.

Decision. . 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 .

Decision. 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 degrees, so the cons disappear. Let's do a reduction at the end.

Example 10: Perform actions .

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

A fraction is the ratio of the numerator to the denominator, and the denominator must not be zero, and the numerator can be any.

When raising any fraction to an arbitrary power, you need to separately raise the numerator and denominator of the fraction to this power, after which we must count these powers and thus get the fraction raised to the power.

For example:

(2/7)^2 = 2^2/7^2 = 4/49

(2 / 3)^3 = (2 / 3) (2 / 3) (2 / 3) = 2^3 / 3^3

negative power

If we are dealing with a negative degree, then we must first “Reverse the fraction”, and only then raise it to a power according to the rule written above.

(2/7)^(-2) = (7/2)^2 = 7^2/2^2

Letter degree

When working with literal values ​​such as "x" and "y", exponentiation follows the same rule as before.

We can also check ourselves by raising the fraction ½ to the 3rd power, as a result we get ½ * ½ * ½ = 1/8 which is essentially the same as

Literal exponentiation x^y

Multiplication and division of fractions with powers

If we multiply powers with the same base, then the base itself remains the same, and we add the exponents. If we divide powers with the same base, then the base of the degree also remains the same, and the exponents are subtracted.

This can be shown very easily with an example:

(3^23)*(3^8)=3^(23+8) = 3^31

(2^4)/(2^3) = 2^(4-3) = 2^1 = 2

We could get the same thing if we simply raised the denominator and numerator separately to the power of 3 and 4, respectively.

Raising a fraction with a power to another power

When raising a fraction, which is already in a power, once again into a power, we must first do the internal exponentiation and then move on to the external part of the exponentiation. In other words, we can simply simply multiply these powers and raise the fraction to the resulting power.

For example:

(2^4)^2 = 2^ 4 2 = 2^8

Uniting, square root

Also, we must not forget that raising absolutely any fraction to the zero power will give us 1, just like any other number, when raised to a power equal to zero, we will get 1.

The usual square root can also be represented as a power of a fraction

Square root 3 = 3^(1/2)

If we are dealing with a square root under which there is a fraction, then we can represent this fraction in the numerator of which there will be a square root of 2 - degrees (because the square root)

And the denominator will also contain the square root, i.e. in other words, we will see the ratio of two roots, this may be useful for solving some problems and examples.

If we raise a fraction that is under the square root to the second power, then we get the same fraction.

The product of two fractions under the same degree will be equal to the product of these two fractions, each of which individually will be under its own degree.

Remember: you can't divide by zero!

Also, do not forget about a very important remark for a fraction such as the denominator should not be equal to zero. In the future, in many equations, we will use this restriction, called ODZ - the range of permissible values

When comparing two fractions with the same base but different degrees, the larger fraction will be the fraction in which the degree will be greater, and the smaller one in which the degree will be less, if not only the bases are equal, but also the degrees, the fraction is considered the same.

Sometimes in mathematics it is necessary to raise a number to a power that represents a fraction. Our article will tell you how to raise a number to a fractional power, and you will see that it is very simple.

A fractional number is very rarely an integer. Often the result of such an erection can be represented with a certain degree of accuracy. Therefore, if the accuracy of the calculation is not specified, then those values ​​are found that are calculated with an accuracy of up to integers, and those that have a large number of signs after the decimal point, leave with roots. For example, the cube root of seven or the square root of two. In physics, the calculated values ​​of these roots are rounded to hundredths when no other degree of accuracy is needed.

Solution algorithm

1. Converting a fractional indicator to an incorrect or proper fraction. Part improper fraction, which is an integer, should not be selected. If a fractional power is represented as an integer and a fractional part, then it must be converted to an improper fraction
2. We calculate the value of the power of a given number, which is equal to the numerator of a proper or improper fraction
3. We calculate the root of the number obtained in paragraph 2, the indicator of which we take the denominator of our fraction

Let us give examples of such calculations

Also, for these calculations, you can download a calculator to your computer or use online calculators, which are very numerous on the Internet, for example.