# How do you rewrite this expression in simplest radical form: (4x^3y^4)^(3/2)?

Mar 23, 2018

The simplified version is $8 {y}^{6} {x}^{4} \sqrt{x}$

#### Explanation:

When you take an exponent of an exponent, it's equal to multiplying both exponents together. Using this tip, we can distribute the $\frac{3}{2}$ exponent as follows:

NOTE: When I color something $\textcolor{red}{\text{red}}$ it means it's in the simplest form.

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

$= {4}^{\frac{3}{2}} \times {x}^{\frac{9}{2}} \times {y}^{\frac{12}{2}}$

$= {4}^{\frac{3}{2}} \times {x}^{\frac{9}{2}} \times \textcolor{red}{{y}^{6}}$

Next, lets simplify the first two terms:

${4}^{\frac{3}{2}} = {\left({4}^{\frac{1}{2}}\right)}^{3} = {\left(\sqrt{4}\right)}^{3} = {2}^{3} = \textcolor{red}{8}$

${x}^{\frac{9}{2}} = {x}^{4 \frac{1}{2}} = \textcolor{red}{{x}^{4} \sqrt{x}}$

Finally, let's reassemble:

${\left(4 {x}^{3} {y}^{4}\right)}^{\frac{3}{2}} = \textcolor{red}{8 {y}^{6} {x}^{4} \sqrt{x}}$