Summarize:

You can also expect to face cube roots in the denominator at some point, though they are rarer. This method also generalizes to roots of any index. 333{\displaystyle {\frac {3}{\sqrt[{3}]{3}}}} Finding an expression that will rationalize the denominator here will be a bit different because we cannot simply multiply by the radical. 331/3{\displaystyle {\frac {3}{3^{1/3}}}} In our case, we are dealing with a cube root, so multiply by 32/332/3.{\displaystyle {\frac {3^{2/3}}{3^{2/3}}}.} Remember that exponents turn a multiplication problem into an addition problem by the property abac=ab+c.{\displaystyle a^{b}a^{c}=a^{b+c}.}  331/3⋅32/332/3{\displaystyle {\frac {3}{3^{1/3}}}\cdot {\frac {3^{2/3}}{3^{2/3}}}} This can generalize to nth roots in the denominator. If we have 1a1/n,{\displaystyle {\frac {1}{a^{1/n}}},} we multiply the top and bottom by a1−1n.{\displaystyle a^{1-{\frac {1}{n}}}.} This will make the exponent in the denominator 1. 331/3⋅32/332/3=32/3{\displaystyle {\frac {3}{3^{1/3}}}\cdot {\frac {3^{2/3}}{3^{2/3}}}=3^{2/3}} If you need to write it in radical form, factor out the 1/3.{\displaystyle 1/3.} 32/3=(32)1/3=93{\displaystyle 3^{2/3}=(3^{2})^{1/3}={\sqrt[{3}]{9}}}
Examine the fraction. Rewrite the denominator in terms of exponents. Multiply the top and bottom by something that makes the exponent in the denominator 1. Simplify as needed.