Question

Assuming that x, y, z are positive real numbers, simplify the following:

`(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))`

Answer 1

We have to simplify the following, assuming that x, y, z are positive real numbers

Given `(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))`

`=(x^(1/2))^(-2/3)(y^4)^(1/2)div(x xxy^(-1/2))^(1/2)`

`=(x^(1/2xx-2/3)xxy^(4xx1/2))/(x^(1/2)xxy^(-1/2xx1/2))`

`=(x^(-1/3)xxy^2)/(x^(1/2)xxy^(-1/4))`

By using the law of rational exponents, `a^mdiva^n=a^(m-n)` we have

`=x^(-1/3-1/2)xxy^(2+1/4)`

`=x^(-5/6)xxy^(9/4)`

`=1/x^(5/6)xxy^(9/4)`

`=y^(9/4)/x^(5/6)`