Question

How do you evaluate `(\frac { x ^ { - 1/ 4} x ^ { 1/ 6} } { x ^ { 1/ 4} x ^ { - 1/ 2} } ) ^ { - 1/ 3}`?

Answer 1

`x^(-1/18) = root(18)x`

Explanation:

Remember that `x^a * x^b = x^(a + b)`

...you can apply this to the numerator and denominator, giving:

`((x^(1/6-1/4))/(x^(1/4 - 1/2)))^(-1/3)`

`=(x^(-1/12)/(x^(-1/4)))^(-1/3)`

Next, remember that `1/x^a = x^(-a)`

...and furthermore, `1/x^(-a) = 1/(1/x^(a)) = x^a`

So with this in mind, you can flip your numerator/denominator upside down, and write:

`= (x^(1/4)/x^(1/12))^(-1/3)`

...and, since `x^a/x^b = x^(a-b)`, you can rewrite this as:

`(x^(1/4 - 1/12))^(-1/3)`

`= (x^(1/6))^(-1/3)`

...and furthermore, remember that `(x^a)^b = x^(a * b)`

So you can rewrite it as:

`= x^(1/6 * -1/3) = x^(-1/18)`

...alternately, this could be written as `1/root(18)x`

...GOOD LUCK!