Assuming that `x , y , z` are positive real numbers, simplify each of the following: `(i)\ (sqrt(x^(-3)))^5` (ii) `(sqrt(x))^(-2/3)\ \ sqrt(y^4)\ -:\ sqrt(x y^(-1/2))`

Explanation

(i) `(sqrt(x^(−3)))^5`
`=(x^(−3/2))^5`
`=x^((−3/2)×5)` [`∵(x^m)^n=x^(mn)`]
`=x^(−15/2)`
`=1/x^(15/2)` [`∵x^(−m)=1/x^m`]
(ii) `(sqrtx)^(−2/3)sqrt(y^4)÷sqrtxy^(−1/2)`
`=(x^(1/2))^(−2/3)(y^(1/2))^4÷x^(1/2)(y^(−1/2))^(1/2)`
`=x^((1/2)×(−2/3))*y^((1/2)×4)÷x^(1/2)*y^((−1/2)×1/2)`
`=x^(−1/3)y^2 ÷x^(1/2)*y^(−1/4)`
`=x^((−1/3)−(1/2))*y^(2-(−1/4))`
`=x^((−2−3)/6)*y^(2+1/4)`
`=x^(−5/6)*y^(9/4)`
`=y^(9/4)/x^(5/6)`.