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Question
Simplify:
`(sqrt2/5)^8div(sqrt2/5)^13`
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Solution
Given `(sqrt2/5)^8div(sqrt2/5)^13`
`(sqrt2/5)^8div(sqrt2/5)^13=(2^(1/2xx8)/5^8)div(2^(1/2xx13)/5^13)`
`=(2^4/5^8)div(2^(13/2)/5^13)`
`=(2^4/5^8)/(2^(13/2)/5^13)`
`=(2^4/5^8)xx(5^13/2^(13/2))`
`=(5^13/5^8)xx(2^4/2^(13/2))`
By using the law of rational exponents `a^m/a^n=a^(m-n)`
`rArr(sqrt2/5)^8div(sqrt2/5)^13=5^(13-8)xx2^(4-13/2)`
`rArr(sqrt2/5)^8div(sqrt2/5)^13=5^5xx2^((4xx2)/(1xx2)-13/2)`
`=5^5xx2^(-5/2)`
`=5^5/2^(5/2)`
`=5^5/root2(2xx2xx2xx2xx2)`
`=5^5/(4sqrt2)`
Hence the value of `(sqrt2/5)^8div(sqrt2/5)^13` is `5^5/(4sqrt2)`
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