# cube root of 9 divided by the cube root of 11, Rationalizing the Denominator

by Kathy
(PA)

how would you solve? cube root of 9 divided by the cube root of 11?

### Comments for cube root of 9 divided by the cube root of 11, Rationalizing the Denominator

 Jun 20, 2011 Rationalizing the Denominator Containing a Cube Root by: Staff The question: by Kathy (PA) how would you solve? cube root of 9 divided by the cube root of 11? The answer: Rationalizing the Denominator is the standard way of simplifying fractions containing radicals in the denominator. Rationalizing the denominator means to “rewrite the fraction so there are no radicals in the denominator”. Your problem as it now stands: = ∛(9) / ∛(11) “IF” the denominator could by multiplied by itself twice [∛(11) * ∛(11)], then the ∛ sign in the denominator would disappear, since: ∛(11) * ∛(11) * ∛(11) = ∛(11*11*11) = ∛(11³) = 11 However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: [∛(11) * ∛(11)]. To apply this concept, multiply the original fraction by [∛(11) * ∛(11)]/ [∛(11) * ∛(11)]. The fraction [∛(11) * ∛(11)]/ [∛(11) * ∛(11)] is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, its value has not been changed. = [original fraction] = [original fraction] * [∛(11) * ∛(11)]/ [∛(11) * ∛(11)] = [original fraction] * 1 = [original fraction] Therefore, = [original fraction] * [∛(11) * ∛(11)]/ [∛(11) * ∛(11)] = {∛(9) / ∛(11)} * {[∛(11) * ∛(11)]/ [∛(11) * ∛(11)]} Multiply both numerators and multiply both denominators, just as you would when multiplying any two fractions: = [∛(9) * ∛(11) * ∛(11)] / [∛(11) * ∛(11) * ∛(11)] = [∛(9) * ∛(11) * ∛(11)] / [∛(11*11*11)] = [∛(9) * ∛(11) * ∛(11)] / [∛(11³)] = [∛(9) * ∛(11) * ∛(11)] / ∛(11³) = [∛(9*11*11)] / 11 = [∛(1089)] / 11 The final answer is: = [∛(1089)] / 11 Check this answer against the original expression with a calculator: [∛(1089)] / 11 = 0.935298 ∛(9) / ∛(11) = 0.935298 Thanks for writing. Staff www.solving-math-problems.com