# How do you simplify (1/9)^(-3/2)?

Oct 28, 2017

Using the formula ${a}^{- b} = \left(\frac{1}{a} ^ b\right)$, and the formula ${\left({a}^{m}\right)}^{n} = {a}^{m n}$, we obtain answer is 27.

#### Explanation:

We have, in our problem,

${\left(\frac{1}{9}\right)}^{- \frac{3}{2}}$ ( Expression 1 )

Rewrite the BASE $\left(\frac{1}{9}\right)$ in our ( Expression 1 ), using the formula ${a}^{- b} = \left(\frac{1}{a} ^ b\right)$, as

${\left({3}^{- 2}\right)}^{- \frac{3}{2}}$ ( Expression 2 )

--------------- Observe that ${3}^{2}$ is 9 and ${3}^{- 2}$ is $\frac{1}{9}$ ----------------

Using the formula ${\left({a}^{m}\right)}^{n} = {a}^{m n}$, we can rewrite our ( Expression 2 ) as

${3}^{\left(- 2\right) \cdot \left(- \frac{3}{2}\right)}$

Simplify by multiplying (-2) by (-3/2) in the Exponent. We get, 3.

Hence, our ( Expression 2 ) can be written as

${3}^{3}$

Which gives us 27.

Hence, ( Expression 1 ) when simplified yields 27.