Simplify: `(i)\ ((sqrt(2))/5)^8\ -:((sqrt(2))/5)^(13)` `(ii)\ ((5^(-1)xx\ 7^2)/(5^2\ xx\ \ 7^(-4)))^(7/2)\ xx\ ((5^(-2)xx\ 7^3)/(5^3xx\ 7^(-5)))^(-5/2)`

Explanation

(i) `((sqrt(2))/5)^8\ -:((sqrt(2))/5)^(13)`
`=((sqrt(2))/5)^8xx(5/sqrt(2))^13`
`=(2^(1/2)/5)^8xx(5/2^(1/2))^13`
`=(2^(8/2)/5^8)xx(5^13/2^(13/2))`
`=(2^4/5^8)xx((5^8xx5^5)/(2^4xx2^(5/2)))`
`=5^5/(2^4sqrt(2)`

(ii) `((5^(-1)xx\ 7^2)/(5^2\ xx\ \ 7^(-4)))^(7/2)\ xx\ ((5^(-2)xx\ 7^3)/(5^3xx\ 7^(-5)))^(-5/2)`
`=((7^2xx7^4)/(5^2xx5))^(7/2)xx((5^3xx5^2)/(7^3xx7^5))^(5/2`
`=(7^6/5^3)^(7/2)xx(5^5/7^8)^(5/2)`
`=(7^21/5^(21/2))xx(5^(25/2)/7^(20))`
`=7^(21-20)xx5^(25/2-21/2)`
`=7xx5^2`
`=175`