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

1 Answer
Sep 15, 2017

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!