Power of a Quotient Property

 

In mathematics, the power of quotient property is similar as power of product property. The power of product property is for multiplying the exponents whereas the power of quotient property is for dividing the exponents. This property helps to solve the problems in simple way. We are going to study the power of quotient property.

 

Explanation to Power of Quotient Property:

If power of different numbers is same then we can write the power as common. In general a power of quotient property has the following format.
`(x/y)^a ` = `(x^a)/(y^a) ` where x and y are not equal to zero.
By studying the power of quotient property we can simplify the problems easily.
For example, `(4/3)^2` = `4^2/3^2`Example Problems to Power of Quotient Property:
Example: 1
Simplify the following problem using power of quotient property.
`(15/5)^3`
Solution:
Given:` (15/5)^3`
By power of quotient property, we can write the given problem as,
`(15/5)^3` = `(15^3)/(5^3)`
Step 1:
15^3 = 5 . 3 . 5 . 3 . 5 . 3
5^3 = 5 . 5 . 5
Step 2:
`(15/5)^3` = `(5 . 3 . 5 . 3 . 5 . 3)/(5 . 5 . 5)`
Cancel the common number 5.
`(15/5)^3 ` = 3^3
Answer: 3^3
Example: 2
Find the answer of the following problem using power of quotient property.
`(4/7)^2`
Solution:
Given: `(4/7)^2`
Step 1:
By power of product property we can write the common power to elements 4 and 7.
`(4/7)^2 ` = `(4^2)/(7^2)`
= `(4 . 4)/(7 . 7)`
= `16/49`
Answer: `16/49`
Example: 3
Simplify the following problem using power of quotient property.
`(10/4)^2`
Solution:
Given: `(10/4)^2`
By power of quotient property, we can write the given problem as,
`(10/4)^2` = `(10^2)/(4^2)`
Step 1:
10^3 = 10 . 10 = 2 .5 . 2 . 5
4^2 = 4 . 4 = 2 . 2 . 2 . 2
Step 2:
`(10/4)^2 ` = `(2 . 5 . 2 . 5)/( 2 . 2 . 2 . 2)`
Cancel the common number 2.
`(10/4)^2` = `(5 . 5)/(2 . 2)`
= `(5^2)/(2^2)`
= `(5/2)^2`
Answer: `(5/2)^2`Practice Problems to Power of Quotient Property:

 


Problem: 1
Simplify the following problem using power of quotient property.
`(2/6)^2`
Answer: `1/9`
Problem: 2
Simplify the following problem using power of quotient property.
`((6 a)/b)^3`
Answer: `(216 a^3)/b^3`