How to multiply two different numbers with different powers

Question

How do you multiply and simplify: $\left(\frac{2}{3}\right)^{1/6}\cdot 18^{1/3}$?

Simplify in surd form.

Answer 1

$$\left( \frac 23\right)^{1/6} * 18^{1/3} = \left(\frac 23\right)^{1/3 * 1/2}*\left((18^2)^{1/2} \right)^{1/3}=\left(\frac 23 * 18^2 \right)^{1/3 *1/2}=(216)^{1/6}$$

Answer 2

If you see that: $2/6=1/3$

Your expression is the following: $$ \begin{equation} \begin{split} \left(\dfrac{2}{3}\right)^{\dfrac{1}{6}}\times\left(18\right)^{\dfrac{1}{3}}&=\left(\dfrac{2}{3}\right)^{\dfrac{1}{6}}\times\left(18\right)^{2\dfrac{1}{6}}\\&=\left(\dfrac{2}{3}\right)^{\dfrac{1}{6}}\times\left(18^2\right)^{\dfrac{1}{6}}\\&=\left(\dfrac{2}{3}\times18^2\right)^{\dfrac{1}{6}}\\&=\left(2\times6\times18\right)^{\dfrac{1}{6}}\\&=\left(216\right)^{\dfrac{1}{6}} \end{split} \end{equation} $$

Answer 3

Try: $$\large\left(\frac{2}{3} \right)^{\frac{1}{6}} \cdot 18^{\frac{1}{3}}=\left(\frac{2}{3} \right)^{\frac{1}{6}} \cdot ((18)^2)^{\frac{1}{6}}=(216^{\frac{1}{6}})\approx\boxed{2.45}$$

Answer 4

All of the other answers in this thread are correct, but oddly they all stop short of noticing that $216=6^3$, and therefore $\sqrt[6]{216}$ can be simplified to $\sqrt{6}$.