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^{(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^{(4)/(4)}\cdot a^{(4)/(4)}\cdot b^{(3)/(4)} \end{gathered}`

Simplifying:

`\sqrt[4]{16a^4b^3}=2ab^{(3)/(4)}`