# 1.4: Exponents

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

The Laws of Exponents let you rewrite algebraic expressions that involve exponents. The last three listed here are really definitions rather than rules.

## Laws of Exponents

All variables here represent real numbers and all variables in denominators are nonzero.

1. $$x^a\cdot x^b=x^{a+b}$$
2. $$\dfrac{x^a}{x^b}=x^{a-b}$$
3. $$\left(x^a\right)^b=x^{ab}$$
4. $$(xy)^a=x^a y^a$$
5. $$\left(\dfrac{x}{y}\right)^b=\dfrac{x^b}{y^b}$$
6. $$x^0=1$$, provided $$x\neq 0$$. [Although in some contexts $$0^0$$ is still defined to be 1.]
7. $$x^{-n}=\dfrac{1}{x^n}$$, provided $$x\neq 0$$.
8. $$x^{1/n}=\sqrt[n]{x}$$, provided $$x\neq 0$$.

## Example $$\PageIndex{1}$$

Simplify $$\left(2x^2\right)^3(4x)$$.

###### Solution

We'll begin by simplifying the $$\left(2x^2\right)^3$$ portion. Using Property 4, we can write

 $$2^3\left(x^2\right)^3(4x)$$ $$8x^6(4x)$$ Evaluate $$2^3$$, and use Property 3. $$32x^7$$ Multiply the constants, and use Property 1, recalling $$x = x^1$$.

Being able to work with negative and fractional exponents will be very important later in this course.

## Example $$\PageIndex{2}$$

Rewrite $$\dfrac{5}{x^3}$$ using negative exponents.

###### Solution

Since $$x^{-n}=\dfrac{1}{x^n}$$, then $$x^{-3}=\dfrac{1}{x^3}$$ and thus $\dfrac{5}{x^3}=5x^{-3}.\nonumber$

## Example $$\PageIndex{3}$$

Simplify $$\left(\dfrac{x^{-2}}{y^{-3}}\right)^2$$ as much as possible and write your answer using only positive exponents.

###### Solution

\begin{align*} \left(\dfrac{x^{-2}}{y^{-3}}\right)^2 & = \dfrac{\left(x^{-2}\right)^2}{\left(y^{-3}\right)^2}\\ & = \dfrac{x^{-4}}{y^{-6}}\\ & = \dfrac{y^6}{x^4} \end{align*}

## Example $$\PageIndex{4}$$

Rewrite $$4\sqrt{x}-\dfrac{3}{\sqrt{x}}$$ using exponents.

###### Solution

A square root is a radical with index of two. In other words, $$\sqrt{x}=\sqrt[2]{x}$$. Using the exponent rule above, $$\sqrt{x}=\sqrt[2]{x}=x^{1/2}$$. Rewriting the square roots using the fractional exponent, $4\sqrt{x}-\dfrac{3}{\sqrt{x}}=4x^{1/2}-\dfrac{3}{x^{1/2}}.\nonumber$

Now we can use the negative exponent rule to rewrite the second term in the expression:

$4x^{1/2}-\dfrac{3}{x^{1/2}}=4x^{1/2}-3x^{-1/2}.\nonumber$

## Example $$\PageIndex{5}$$

Rewrite $$\left(\sqrt{p^5}\right)^{-1/3}$$ using only positive exponents.

###### Solution

\begin{align*} \left(\sqrt{p^5}\right)^{-1/3} & = \left(\left(p^5\right)^{1/2}\right)^{-1/3}\\ & = p^{-5/6}\\ & = \frac{1}{ p^{5/6}} \end{align*}

## Example $$\PageIndex{6}$$

Rewrite $$x^{-4/3}$$as a radical.

###### Solution

\begin{align*} x^{-4/3} & = \frac{1}{x^{4/3}} \\ & = \frac{1}{\left(x^{1/3}\right)^4} \quad \text{(since $$\frac{4}{3}=4\cdot\frac{1}{3}$$)}\\ & = \frac{1}{\left(\sqrt[3]{x}\right)^4} \quad \text{(using the radical equivalence)} \end{align*}

This page titled 1.4: Exponents is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Shana Calaway, Dale Hoffman, & David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.