# How do you simplify (4/81)^(3/2)?

Oct 5, 2016

${\left(\frac{4}{81}\right)}^{\frac{3}{2}} = \frac{8}{729}$

#### Explanation:

We use here the identities ${\left(\frac{a}{b}\right)}^{m} = {a}^{m} / {b}^{m}$ and ${\left({a}^{m}\right)}^{n} = {a}^{\left(m \times n\right)}$

Hence ${\left(\frac{4}{81}\right)}^{\frac{3}{2}}$

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

= ${\left({2}^{2} / {3}^{4}\right)}^{\frac{3}{2}}$

= ${\left({2}^{2}\right)}^{\frac{3}{2}} / {\left({3}^{4}\right)}^{\frac{3}{2}}$ - using ${\left(\frac{a}{b}\right)}^{m} = {a}^{m} / {b}^{m}$

= 2^((2xx3/2))/(3^((4xx3/2)) - using ${\left({a}^{m}\right)}^{n} = {a}^{\left(m \times n\right)}$

= ${2}^{3} / {3}^{6}$

= $\frac{2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3 \times 3}$

= $\frac{8}{729}$