Division of exponents
Definition of Quotient Property of Exponents
The basic rule for divtopicIding exponents with the same base is that we subtract the given powers or indices. By using the laws of exponents, it becomes simple to solve the division and multiplication of exponents. This technique is also referred to as the Quotient Property of Exponents.
For example: 9^6 / 9^4 = 9^(6-4) = 9^2= 9*9= 81
Here ^ means raise to the power.
In this article, you will learn rules for the division of positive exponents, negative exponents, fractional exponents, and division of exponents with coefficients.
Rules for division of exponents:
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When the base is the same and the powers are different.
x^a / x ^b = x^( a-b)
Here the base is x. Powers are different ie a and b. So, here subtract
the power and write its answer on the base.
Example: (4^4) / (4^2) = 4 ^ (4-2) = 4 ^ 2 = 8
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When the base is different and powers are the same.
x^a / y^a = (x/y)^a
Here, x and y are two different bases and their power is the same i.e. a. So, we
will divtopicIde x and y. They will share the same power.
Example: (4^2) / (5^2) = (⅘)^2 = 16/25
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When the base and the powers both are different
x^a / y^b
Here, the bases are x and y. Their powers are a and b respectively.
So, simply we will solve it. Both are exponents and divtopicIde its answer with
each other.
Example: (4^2) / (2^3) = 16/8 = 2
Practise problems:
Example 1: DivtopicIde 10^7 by 10^4
= 10 ^ (7-4)
= 10^3
= 10 * 10 * 10= 1000
Example 2: DivtopicIde 8^ 5 by 4^ 5
= (8/4)^5
= 2^5 = 2*2*2*2*2 = 32
Example 3: DivtopicIde 3^3 / 9^ 2
= 3^3 /9^2
= 3* 3* 3 / 9*9*9
= 1/27
Division of negative exponent:
If the power is negative in the division of exponents, then we will take the reciprocal of the base. This will make the power positive.
Example: (4^(-3)) = (1 / 4)^3 = 1/64
Practise problems:
Example 2: DivtopicIde 6^-9 and 6^ -5
= (1/ 6)^9 / (1/ 6) ^ 5
= (1/ 6) 9-5
= (1/ 6) 4
Division of fractional exponent:
Whenever a fractional exponent is there we need to subtract the exponent or powers of the bases.
For example : 5^½ / 5^½
= 5^ (½ - ½ )
= 5^ 0 = 1
As per the law of exponents, any number raised to 0 is always one.
Practice problems:
Example 3: 25^½ / 25^¼
= 25^ (½ + ¼ )
= 25 ^ ¾ (here we have solved the fraction by taking lcm of denominator
2 &4.)
Now as 25 is square of five= 5^2
=5^2* 3/ 4 ( we have simplified the powers ie 2* 3/ 4 = 2*3/ 4= 3/2
= 5^3/2
Division of Exponents having Coefficients
We must divtopicIde expressions with coefficients in various circumstances. These coefficients, which are associated with their bases, divtopicIded in the same manner that any normal fraction could be divtopicIded. It should be remembered that the coefficients may be divtopicIded even though the bases of the expressions vary.
Example:
DivtopicIde 16 x (a^6) by 4 x (a^3)
Solution: To divtopicIde expressions with coefficients, apply the procedures below. The coefficients in this example are 16 and 4, and the remaining are variables.
We begin by rewriting the expression as a fraction, 16(a^6) / 4(a^3)
The coefficients are then divtopicIded, resulting in 16/4=4
After that, we may solve the variable using the quotient property of exponents, which is a^6/a^3= a^(6 - 3) = a^3
As a result, the final solution is 4(a^3).
