Simplify $\sqrt[3]{162x^6y^7}$

Question

The answer is $3x^2 y^2 \sqrt[3]{6y}$

How does $\sqrt[3]{162x^6y^7}$ equal $3x^2 y^2\sqrt[3]{6y}$?

Answer 1

\begin{align} \sqrt[3]{162x^6y^7}&=\sqrt[3]{27x^6y^6*6y}\\ &=\sqrt[3]{27x^6y^6}\times\sqrt[3]{6y}\\ &=3x^2y^2\sqrt[3]{6y} \end{align}

Answer 2

$162x^6y^7=(3x^2y^2)^3\cdot6y$, such that

$$\sqrt[3]{162x^6y^7}=\sqrt[3]{(3x^2y^2)^3\cdot6y}=\sqrt[3]{(3x^2y^2)^3}\sqrt[3]{6y}=3x^2y^2\sqrt[3]{6y}$$

Answer 3

First note that 162 factors as $2\cdot 3^4$. So collecting our powers of 3 reveals $$\sqrt[3]{162x^6y^7} = \sqrt[3]{3^3 \cdot (x^3)^2\cdot (y^3)^2 \cdot (2\cdot 3 \cdot y)} = 3x^2y^2\sqrt[3]{6y}.$$