Question

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

$\frac{1}{\sqrt{3}}$

Short answer

The radical expression $\frac{1}{\sqrt{3}}$can be written$3^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as using exponents.

Step-by-step solution

Step 1. Definition of nth root and rational exponents.

• If n is a positive integer, then the principal$n^{th}$root of ais defined as follows

$\sqrt[n]{a}=b\iff a=b^n$

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

• $a^{-n}=\frac{1}{a^n}$
  

Step 3. Evaluate the given expression.

Given$\frac{1}{\sqrt{3}}$

Here the exponent of radical in the denominator is$\frac{1}{2}$ and in the denominator, so we will reciprocate it to write in the exponent.

Step 4. Simplify the given expression.

$\begin{array}{l}\frac{1}{\sqrt{3}}=\frac{1}{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\\ \Rightarrow \frac{1}{\sqrt{3}}=3^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{................}\left[a^{-n}=\frac{1}{a^n}\right]\end{array}$

So, the radical expression$\frac{1}{\sqrt{3}}$ can be written$3^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as using exponents.