Question

Explain what $4^{\frac{3}{2}}$means, then calculate $4^{\frac{3}{2}}$in two different ways:

${\left(4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^x$= ______

${\left(4^3\right)}^x$= _______

Short answer

$4^{\frac{3}{2}}$means 4 is the base and $\frac{3}{2}$is the exponent.

$4^{\frac{3}{2}}={\left(\sqrt[2]{4}\right)}^3=\sqrt[2]{4^3}$

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 a is 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 exponentrole="math" localid="1643262022097" $\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$in lowest terms, where m and n are integers and$n>0$, we definerole="math" localid="1643262066043" $a^{\raisebox{1ex}{$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.

$a^{mn}={\left(a^m\right)}^n$

Step 3. Analyze the given value.

Given$4^{\frac{3}{2}}$

We will evaluate it using a rational exponent.

The given expression means 4 is the base and$\frac{3}{2}$ is the exponent.

Step 4. Calculate the given expression.

Using rational exponent and laws of exponents, we can write$4^{\frac{3}{2}}$ as:

$\begin{array}{l}{4}^{\frac{3}{2}}=4^{\frac{1}{2}\cdot 3}\\ \Rightarrow 4^{\frac{3}{2}}={\left(4^{\frac{1}{2}}\right)}^3\\ \Rightarrow 4^{\frac{3}{2}}={\left(\sqrt[2]{4}\right)}^3\end{array}$

or

$\begin{array}{l}{4}^{\frac{3}{2}}=4^{3\cdot \frac{1}{2}}\\ \Rightarrow 4^{\frac{3}{2}}={\left(4^3\right)}^{\frac{1}{2}}\\ \Rightarrow 4^{\frac{3}{2}}=\sqrt[2]{4^3}\end{array}$