How do you simplify \root[ 3] { \frac { 25x ^ { 11} } { 9w ^ { 2} } }?

1 Answer
Nov 11, 2017

rationalize denominator
(5x^3 root(3)(x^2w))/(3w)

Explanation:

root(3)(25x^11)/root(3)(9w^2)

we need to rationalize the denominator first.

What we are looking for are factors that are multiples of the index, or perfect cubes.

root(3)(25x^11)/root(3)(9w^2)

9 * 3 = 27 meaning 3^3 is 27

w^2 *w = w^3 giving us another perfect cube rationalizing the denominator

root(3)(25x^11)/root(3)(9w^2)*root(3)(3w)/root(3)(3w) = (root(3)125 root(3)(w)root(3)(x^11))/(root(3)(27w^3)

In the numerator:

root(3)125 = 5 Is a perfect cube

root(3)w is fully simplified

root(3)x^9 * root(3)x^2 = root(3)x^11

x^(9/3) = x^3

root(3)27 root(3) w^3 This is the denominator

root(3)27 = 3
root(3)w^3 = w

put it all back together over a single fraction

(5x^3 root(3)(x^2)root(3)w)/(3w)