# Simplify the following? (x^(5/4) sqrt(16x^3)) /(8x)^(2/3)

${x}^{\frac{25}{12}}$

#### Explanation:

I think the question here is to simplify the following:

$\frac{{x}^{\frac{5}{4}} \sqrt{16 {x}^{3}}}{8 x} ^ \left(\frac{2}{3}\right)$

I'll first move the term out of the denominator. Remember that $\frac{1}{x} = {x}^{- 1}$, so we can rewrite the denominator:

$\frac{1}{8 x} ^ \left(\frac{2}{3}\right) = {\left(8 x\right)}^{- \frac{2}{3}}$

and we can simplify further:

${\left(8 x\right)}^{- \frac{2}{3}} = {8}^{- \frac{2}{3}} {x}^{- \frac{2}{3}} = {\left({2}^{3}\right)}^{- \frac{2}{3}} {x}^{- \frac{2}{3}} = {2}^{3 \times \left(- \frac{2}{3}\right)} {x}^{- \frac{2}{3}} = {2}^{-} 2 {x}^{- \frac{2}{3}}$

We can now write the original as:

$\frac{{x}^{\frac{5}{4}} \sqrt{16 {x}^{3}}}{8 x} ^ \left(\frac{2}{3}\right) = \left({x}^{\frac{5}{4}}\right) \left(\sqrt{16 {x}^{3}}\right) \left({2}^{-} 2 {x}^{- \frac{2}{3}}\right)$

Let's rewrite the middle term:

$\sqrt{16 {x}^{3}} = {16}^{\frac{1}{2}} {x}^{\frac{3}{2}} = 4 {x}^{\frac{3}{2}}$

And substitue in:

$\left({x}^{\frac{5}{4}}\right) \left(4 {x}^{\frac{3}{2}}\right) \left({2}^{-} 2 {x}^{- \frac{2}{3}}\right)$

Let's now do the multiplication. I'll group the constants together and the variables together:

$\left(4\right) \left({2}^{-} 2\right) \times \left({x}^{\frac{5}{4}}\right) \left({x}^{\frac{3}{2}}\right) \left({x}^{- \frac{2}{3}}\right)$

For the constants, we have:

$4 \times {2}^{-} 2 = 4 \times \frac{1}{4} = 1$

For the variables, remember that we can use the rule

${x}^{a} \times {x}^{b} = {x}^{a + b}$

We have:

$\left({x}^{\frac{5}{4}}\right) \left({x}^{\frac{3}{2}}\right) \left({x}^{- \frac{2}{3}}\right) = {x}^{\frac{5}{4} + \frac{3}{2} - \frac{2}{3}} = {x}^{\frac{15}{12} + \frac{18}{12} - \frac{8}{12}} = {x}^{\frac{25}{12}}$