Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.
Number c is n-th power of a number a When:
Operations with degrees.
1. Multiplying degrees with the same base, their indicators add up:
a ma n = a m + n .
2. In the division of degrees with the same base, their indicators are subtracted:
3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:
(abc…) n = a n b n c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n / b n .
5. Raising a power to a power, the exponents are multiplied:
(am) n = a m n .
Each formula above is correct in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
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 dividend and the divisor of the roots:
3. When raising a root to a power, it is enough to raise the root number to this power:
4. If we increase the degree of the root in n once and at the same time raise to n th power is a radical number, then the value of the root will not change:
5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:
Degree with a negative exponent. The degree of a number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:
Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.
Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
A degree with a fractional exponent. To raise a real number A to a degree m/n, you need to extract the root n th degree of m th power of this number A.
It makes sense to talk about operations with algebraic fractions. The following operations are defined with algebraic fractions: addition, subtraction, multiplication, division, and raising to a natural power. Moreover, all these actions are closed, in the sense that as a result of their execution, an algebraic fraction is obtained. Let's analyze each of them in order.
Yes, it is immediately worth noting that operations with algebraic fractions are generalizations of the corresponding operations with ordinary fractions. Therefore, the corresponding rules almost verbatim coincide with the rules for performing addition and subtraction, multiplication, division and raising to a power of ordinary fractions.
Page navigation.
Addition of algebraic fractions
The addition of any algebraic fractions fits one of the two following cases: in the first, fractions with the same denominators are added, in the second, with different ones. Let's start with the rule of adding fractions with the same denominators.
To add algebraic fractions with the same denominators, you need to add the numerators, and leave the denominator the same.
The voiced rule allows you to move from adding algebraic fractions to adding polynomials that are in numerators. For example, .
To add algebraic fractions with different denominators, you need to act according to the following rule: bring them to a common denominator, and then add the resulting fractions with the same denominators.
For example, when adding algebraic fractions and they must first be brought to a common denominator, as a result they will take the form 
And 
respectively, after which the addition of these fractions with the same denominators is performed: .
Subtraction
The next step, the subtraction of algebraic fractions, is performed in the same way as addition. If the denominators of the original algebraic fractions are the same, then you just need to subtract the polynomials in the numerators, and leave the denominator the same. If the denominators are different, then the reduction to a common denominator is performed first, after which the resulting fractions with the same denominators are subtracted.
Let's give examples.
Let's subtract algebraic fractions and , their denominators are the same, therefore . The resulting algebraic fraction can be further reduced: 
.
Now subtract the fraction from the fraction. These are algebraic fractions with different denominators, therefore, first we bring them to a common denominator, which in this case is 5 x (x-1) , we have 
And 
. It remains to do the subtraction: 
Multiplication of algebraic fractions
Algebraic fractions can be multiplied. This action is carried out similarly to the multiplication of ordinary fractions according to the following rule: in order to multiply algebraic fractions, you need to multiply the numerators separately, and separately the denominators.
Let's take an example. Multiply an algebraic fraction by a fraction. According to the stated rule, we have 
. It remains to convert the resulting fraction to an algebraic fraction, for this, in this case, you need to perform the multiplication of a monomial and a polynomial (and in the general case, the multiplication of polynomials) in the numerator and denominator: 
.
It is worth noting that before multiplying algebraic fractions, it is desirable to factorize the polynomials that are in their numerators and denominators. This is due to the possibility of reducing the resulting fraction. For example, 
.
This action is discussed in more detail in the article.
Division
We move on to actions with algebraic fractions. Next in line is the division of algebraic fractions. The following rule reduces the division of algebraic fractions to multiplication: to divide one algebraic fraction by another, you need to multiply the first fraction by the reciprocal of the second.
An algebraic fraction inverse to a given fraction is understood as a fraction with the numerator and denominator rearranged. In other words, two algebraic fractions are considered mutually inverse if their product is identically equal to one (by analogy with).
Let's take an example. Let's do the division 
. The reciprocal of the divisor is . Thus, .
For more detailed information, refer to the article mentioned in the previous paragraph multiplication and division of algebraic fractions.
Raising an algebraic fraction to a power
Finally, we move on to the last action with algebraic fractions - raising to a natural power. , as well as how we defined the multiplication of algebraic fractions, allows us to write down the rule for raising an algebraic fraction to a power: you need to separately raise the numerator to this power, and separately the denominator.
Let's show an example of this action. Let's raise an algebraic fraction to the second power. According to the above rule, we have 
. It remains to raise the monomial in the numerator to a power, and also to raise the polynomial in the denominator to a power, which will give an algebraic fraction of the form 
.
The solution of other characteristic examples is shown in the article raising an algebraic fraction 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 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 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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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 n=1, which means the number A taken once and respectively: $a^n= 1$.
If 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.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.
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 raise the numerator and denominator of the fraction to this power separately, 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 degree
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 go 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 one 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, but also the degrees are equal, the fraction is considered the same.
