((1)/(81))^(1)/(4) Its asking me to put this expression into radical form.

Question

`((1)/(81))^(1)/(4)`

Its asking me to put this expression into radical form.

Answer 1

If
`n>1`, we write
`x^(1/n)=\sqrt[n]x`. So

`\left(\frac1{81}\right)^(1/4)=\sqrt[4]{\frac1{81}}`

We can simplify this further by using properties of square roots:

`√(\frac ab)=(\sqrt a)/(\sqrt b)\implies\sqrt[4]{\frac1{81}}=\frac{\sqrt[4]1}{\sqrt[4]{81}}=\frac1{\sqrt[4]{81}}`

Next,
`81=9^2=(3^2)^2=3^4`, and so
`\sqrt[4]{81}=\sqrt[4]{3^4}=3`, so

`\left(\frac1{81}\right)^(1/4)=\frac13`