Question

Convert the radical to exponential form. Assume variables represent positive real numbers.

Answer 1

Exponential Form of Radicals

A radical can be expressed in exponential form by using the equivalence:

$\sqrt[m]{x^n}=x^{\frac{n}{m}}$

We are given the expression:

$\sqrt[4]{16a^4b^3}$

It can be separated into several radicals:

$\sqrt[4]{16a^4b^3}=\sqrt[4]{16}\cdot\sqrt[4]{a^4}\cdot\sqrt[4]{b^3}$

Now we apply the equivalence on each individual radical:

$\begin{gathered} \sqrt[4]{16a^4b^3}=\sqrt[4]{2^4}\cdot\sqrt[4]{a^4}\cdot\sqrt[4]{b^3} \\ \sqrt[4]{16a^4b^3}=2^{\frac{4}{4}}\cdot a^{\frac{4}{4}}\cdot b^{\frac{3}{4}} \end{gathered}$

Simplifying:

$\sqrt[4]{16a^4b^3}=2ab^{\frac{3}{4}}$