# What is the correct radical form of this expression (32a^10b^(5/2))^(2/5)?

Dec 12, 2016

${\left(32 {a}^{10} {b}^{\frac{5}{2}}\right)}^{\frac{2}{5}} = 4 {a}^{4} b$

#### Explanation:

First, rewrite $32$ as $2 \times 2 \times 2 \times 2 \times 2 = {2}^{5}$:

${\left(32 {a}^{10} {b}^{\frac{5}{2}}\right)}^{\frac{2}{5}} = {\left({2}^{5} {a}^{10} {b}^{\frac{5}{2}}\right)}^{\frac{2}{5}}$

The exponent can be split up by multiplication, that is, ${\left(a b\right)}^{c} = {a}^{c} \cdot {b}^{c}$. This is true for a product of three parts, such as ${\left(a b c\right)}^{d} = {a}^{d} \cdot {b}^{d} \cdot {c}^{d}$. Thus:

${\left({2}^{5} {a}^{10} {b}^{\frac{5}{2}}\right)}^{\frac{2}{5}} = {\left({2}^{5}\right)}^{\frac{2}{5}} \cdot {\left({a}^{10}\right)}^{\frac{2}{5}} \cdot {\left({b}^{\frac{5}{2}}\right)}^{\frac{2}{5}}$

Each of these can be simplified using the rule ${\left({a}^{b}\right)}^{c} = {a}^{b c}$.

${\left({2}^{5}\right)}^{\frac{2}{5}} \cdot {\left({a}^{10}\right)}^{\frac{2}{5}} \cdot {\left({b}^{\frac{5}{2}}\right)}^{\frac{2}{5}} = {2}^{5 \times \frac{2}{5}} \cdot {a}^{10 \times \frac{2}{5}} \cdot {b}^{\frac{5}{2} \times \frac{2}{5}}$

$\textcolor{w h i t e}{{\left({2}^{5}\right)}^{\frac{2}{5}} \cdot {\left({a}^{10}\right)}^{\frac{2}{5}} \cdot {\left({b}^{\frac{5}{2}}\right)}^{\frac{2}{5}}} = {2}^{2} \cdot {a}^{4} \cdot {b}^{1}$

$\textcolor{w h i t e}{{\left({2}^{5}\right)}^{\frac{2}{5}} \cdot {\left({a}^{10}\right)}^{\frac{2}{5}} \cdot {\left({b}^{\frac{5}{2}}\right)}^{\frac{2}{5}}} = 4 {a}^{4} b$