How do you simplify root3 (x^6 y^7)/root3 9?

May 10, 2018

$\frac{\sqrt[3]{{x}^{6} {y}^{7}}}{\sqrt[3]{9}}$ can be written as =${\left(x y\right)}^{2} \sqrt[3]{\frac{y}{9}}$

Explanation:

You have the expression $\frac{\sqrt[3]{{x}^{6} {y}^{7}}}{\sqrt[3]{9}}$
Remember that the cubic root root(3) of something means a number which multiplied with itself three times $\left(z \cdot z \cdot z\right)$ gives the number under the root sign.

("multiplied with itself three times" is, of course, not quite precise, but I mean that n is a factor three times, i.e. $z \cdot z \cdot z$.)

Let's rewrite the expression a little:

$\frac{\sqrt[3]{{x}^{6} {y}^{7}}}{\sqrt[3]{9}} = \sqrt[3]{{\left(x y\right)}^{6} \frac{y}{3} ^ 2}$
=${\left(x y\right)}^{2} \sqrt[3]{\frac{y}{3} ^ 2}$
Or if you will
=${\left(x y\right)}^{2} \sqrt[3]{\frac{y}{9}}$
depending on what you prefer.

We may be fooled to think that $\sqrt[3]{9}$ can be simplified, but it is not a cubic number. If we had had $27 = {3}^{3}$ instead, we would get the nice expression
=${\left(x y\right)}^{2} / 3 \sqrt[3]{y}$