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Question
Simplify:
`((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)`
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Solution
Given `((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)`
`((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)=((5^(-1xx7/2)xx7^(2xx7/2))/(5^(2xx7/2)xx7^(-4xx7/2)))xx((5^(-2xx(-5)/2)xx7^(3xx(-5)/2))/(5^(3xx(-5)/2)xx7^(-5xx(-5)/2)))`
`rArr((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)=(5^((-7)/2)xx7^7)/(5^7xx7^-14)xx(5^5xx7^((-15)/2))/(5^((-15)/2)xx7^(25/2))`
By using the law of rational exponents `a^m/a^n=a^(m-n)` we have
`rArr((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)=(5^((-7)/2)xx7^7)/(5^7xx7^-14)xx(5^5xx7^((-15)/2))/(5^((-15)/2)xx7^(25/2))`
`=5^((-7)/2-7)xx7^(7+14)xx5^(5+15/2)xx7^(-15/2-25/2)`
`=5^((-7)/2-14/2)xx7^21xx5^(10/2+15/2)xx7^(-40/2)`
`=5^(-7/2-14/2+10/2+15/2)xx7^(21-40/2)`
`=5^((-7-14+10+15)/2)xx7^((42-40)/2)`
`=5^(4/2)xx7^(2/2)`
`=5^2xx7^1`
`=25xx7`
= 175
Hence the value of `((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)` is 175
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