How do you write 1/(3sqrt2^-n) as a fractional exponent?

Aug 2, 2015

$\textcolor{red}{\frac{1}{3 {\left(\sqrt{2}\right)}^{- n}} = {2}^{\frac{n}{2}} / 3}$

Explanation:

$\frac{1}{x} ^ \left(- n\right) = {x}^{n}$, so

$\frac{1}{3 {\left(\sqrt{2}\right)}^{- n}} = {\left(\sqrt{2}\right)}^{n} / 3 = {\left({2}^{\frac{1}{2}}\right)}^{n} / 3$

Also, ${\left({x}^{a}\right)}^{b} = {x}^{a b}$, so

${\left({2}^{\frac{1}{2}}\right)}^{n} / 3 = {2}^{\frac{n}{2}} / 3$

$\frac{1}{3 {\left(\sqrt{2}\right)}^{- n}} = {2}^{\frac{n}{2}} / 3$