Question

frac{1}{4}" alt="(\frac{1}{81})^\frac{1}{4}" align="absmiddle" class="latex-formula">

Its asking me to put this expression into radical form.

Answer 1

If $n>1$, we write $x^{1/n}=\sqrt[n]x$. So

$\left(\dfrac1{81}\right)^{1/4}=\sqrt[4]{\dfrac1{81}}$

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

$\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}\implies\sqrt[4]{\dfrac1{81}}=\dfrac{\sqrt[4]1}{\sqrt[4]{81}}=\dfrac1{\sqrt[4]{81}}$

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

$\left(\dfrac1{81}\right)^{1/4}=\dfrac13$