# How do you simplify the following expression?

Mar 24, 2018

See below for the detailed steps.

The simplified form is $4 \sqrt[6]{{x}^{11} {y}^{7}}$

#### Explanation:

We can write a root as a fractional exponent or vice versa, so let's start here:

${\left(4 {x}^{3} y\right)}^{\frac{1}{2}} \cdot {\left(8 x {y}^{2}\right)}^{\frac{1}{3}}$

When we have an exponent on an expression in brackets we mutiply exponents:

$\left({4}^{\frac{1}{2}} {x}^{\frac{3}{2}} {y}^{\frac{1}{2}}\right) \cdot \left({8}^{\frac{1}{3}} {x}^{\frac{1}{3}} {y}^{\frac{2}{3}}\right)$

Now ${4}^{\frac{1}{2}} = 2$ and ${8}^{\frac{1}{3}} = 2$ with that we can collect like terms by multiplying, and to multiply we add exponents:

$\left(2 {x}^{\frac{3}{2}} {y}^{\frac{1}{2}}\right) \cdot \left(2 {x}^{\frac{1}{3}} {y}^{\frac{2}{3}}\right)$

$= 4 {x}^{\left(\frac{3}{2} + \frac{1}{3}\right)} {y}^{\left(\frac{1}{2} + \frac{2}{3}\right)}$

$= 4 {x}^{\left(\frac{9}{6} + \frac{2}{6}\right)} {y}^{\left(\frac{3}{6} + \frac{4}{6}\right)}$

$= 4 {x}^{\left(\frac{11}{6}\right)} {y}^{\left(\frac{7}{6}\right)}$

$= 4 \sqrt[6]{{x}^{11} {y}^{7}}$