Question

Write each radical expression using exponents, and each exponential expression using radicals.

$a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

Short answer

The exponential expression $a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$can be written as $\sqrt[5]{a^2}$using radical.

Step-by-step solution

Step 1. Definition of nth root and rational exponents.

• If n is a positive integer, then the principal${\mathit{n}}^{\mathbf{t}\mathbf{h}}$root of ais defined as follows

$\begin{array}{l}\sqrt[n]{a}=b\iff a=b^n\\ \sqrt[n]{a}=a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}\end{array}$

If n is even, we must have$a\ge 0,b\ge 0$

• For any rational exponent $\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$in lowest terms, where m and n are integers and$n>0$, we define$a^{\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$

If n is even, we must have$a\ge 0$

Step 2. Laws of Exponents.

a and b are real numbers and m and n are integers

• $a^{mn}={\left(a^m\right)}^n$
  

Step 3. Evaluate the given expression.

Given$a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

Here we will first remove the fraction from the exponent to convert it into a radical.

Step 4. Simplify the given expression.

$\begin{array}{l}{a}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}={\left(a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^2\\ \Rightarrow a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}={\left(\sqrt[5]{a}\right)}^2\\ \Rightarrow a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}=\sqrt[5]{a^2}\end{array}$

So, the exponential expression$a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ can be written as$\sqrt[5]{a^2}$ using radical.