# 6.2: Quotient Rules for Exponents

Difficulty Level: At Grade Created by: CK-12

Suppose you have the expression:

\begin{align*}\frac{x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y}{x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y}\end{align*}

How could you write this expression in a more concise way?

### Watch This

James Sousa: Simplify Exponential Expressions- Quotient Rule

### Guidance

In the expression \begin{align*}x^3\end{align*}, the \begin{align*}x\end{align*} is called the base and the \begin{align*}3\end{align*} is called the exponent. Exponents are often referred to as powers. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

• \begin{align*}x^3=x\cdot x \cdot x\end{align*}
• \begin{align*}2^4=2\cdot 2 \cdot 2 \cdot 2=16\end{align*}.

There are many rules that have to do with exponents (often called the Laws of Exponents) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and quotients.

RULE: To divide two powers with the same base, subtract the exponents.

\begin{align*}& \qquad \qquad \ {\color{red} m \ \text{factors}}\\ & \qquad \qquad \qquad {\color{red}\uparrow}\\ & \frac{a^m}{a^n}=\frac{\overleftrightarrow{(a \times a \times \ldots \times a)}}{\underleftrightarrow{(a \times a \times \ldots \times a)}} \ m>n;a \neq 0\\ & \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \ {\color{red} n \ \text{factors}}\\ & \frac{a^m}{a^n}=\underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \ {\color{red} m-n \ \text{factors}}\\ & \frac{a^m}{a^n}=a^{\color{red}m-n}\end{align*}

RULE: To raise a quotient to a power, raise both the numerator and the denominator to the power.

\begin{align*}& \left(\frac{a}{b} \right)^n= \underleftrightarrow{\frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\uparrow}\\ & \left(\frac{a}{b}\right)^n=\frac{\overleftrightarrow{(a \times a \times \ldots \times a)}}{\underleftrightarrow{(b \times b \times \ldots \times b)}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \left(\frac{a}{b} \right)^n=\frac{a^{\color{red}n}}{b^{\color{red}n}} \ (b \neq 0)\end{align*}

#### Example A

Evaluate \begin{align*}2^7 \div 2^3\end{align*}.

Solution:

\begin{align*}& 2^7 \div 2^3 && \text{The base is} \ 2'.\\ & 2^{7-3} && \text{Keep the base of} \ 2' \ \text{and subtract the exponents}.\\ & 2^{\color{red}4} && \text{The answer is in exponential form}.\end{align*}

The answer can be taken one step further. The base is numerical so the term can be evaluated.

\begin{align*}& 2^4 = 2 \times 2 \times 2 \times 2\\ &{\color{red}2^4} = {\color{red}16}\\ & \boxed{2^7 \div 2^3 =2^4=16}\end{align*}

#### Example B

Evaluate \begin{align*}\frac{x^8}{x^2}\end{align*}.

Solution:

\begin{align*}& \frac{x^8}{x^2} && \text{The base is} \ x.\\ & x^{8-2} && \text{Keep the base of} \ x \ \text{and subtract the exponents.}\\ & x^{\color{red}6} && \text{The answer is in exponential form.}\\ & \boxed{\frac{x^8}{x^2}=x^6}\end{align*}

#### Example C

Evaluate \begin{align*}\frac{16x^5 y^5}{4x^2 y^3}\end{align*}.

Solution:

\begin{align*}& \frac{16x^5 y^5}{4x^2 y^3} && \text{The bases are} \ x \ \text{and} \ y.\\ & 4 \left( \frac{x^5 y^5}{x^2 y^3} \right) && \text{Divide the coefficients -} \ 16 \div 4=4. \ \text{Keep the base of} \ x \ \text{and} \ y \ \text{and}\\ & && \text{subtract the exponents of the same base.}\\ & 4x^{5-2}y^{5-3}\\ & 4x^{{\color{red}3}} y^{\color{red}2}\end{align*}

#### Concept Problem Revisited

\begin{align*}\frac{x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y}{x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y}\end{align*} can be rewritten as \begin{align*}\frac{x^9y^5}{x^6y^3}\end{align*} and then simplified to \begin{align*}x^3y^2\end{align*}.

### Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression \begin{align*}2^5\end{align*}, ‘2’ is the base. In the expression \begin{align*}(-3y)^4\end{align*}, ‘\begin{align*}-3y\end{align*}’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression \begin{align*}2^5\end{align*}, ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: \begin{align*}2^5=2 \times 2 \times 2 \times 2 \times 2\end{align*}
In the expression \begin{align*}(-3y)^4\end{align*}, ‘4’ is the exponent. It means to multiply \begin{align*}-3y\end{align*} times itself 4 times as shown here: \begin{align*}(-3y)^4=-3y \times -3y \times -3y \times -3y\end{align*}.
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

### Guided Practice

Evaluate each of the following.

1. \begin{align*}\left(\frac{2}{3}\right)^2\end{align*}

2. \begin{align*}\left(\frac{x}{6}\right)^3\end{align*}

3. \begin{align*}\left(\frac{3x^2}{4y^3}\right)^2\end{align*}

4. \begin{align*}\left(\frac{4a^5b^3}{6ab}\right)^3\end{align*}

1.

\begin{align*}& \left(\frac{2}{3}\right)^2 && \text{The base is} \ \frac{2}{3}'.\\ & \frac{2^{1 \times 2}}{3^{1 \times 2}} && \text{Keep the base of} \ \frac{2}{3}' \ \text{and multiply the exponents of both the numerator and the}\\ & && \text{denominator by} \ 2.\\ & \frac{2^{\color{red}2}}{3^{\color{red}2}} && \text{The answer is in exponential form}.\end{align*}
The answer can be taken one step further. The base is numerical so each term can be evaluated.
\begin{align*}& 2^2 = 2 \times 2 \qquad \ 3^2=3 \times 3\\ & 2^2=4 \qquad \qquad 3^2=9\\ & \frac{{\color{red}4}}{{\color{red}9}}\\ & \boxed{\left(\frac{2}{3}\right)^2=\frac{2^2}{3^2}=\frac{4}{9}}\end{align*}

2.

\begin{align*}& \left(\frac{x}{6}\right)^3 && \text{The base is} \ \frac{x}{6}'.\\ & \frac{x^{1 \times 3}}{6^{1 \times 3}} && \text{Keep the base of} \ \frac{x}{6}' \ \text{and multiply the exponents of both the numerator and the}\\ & && \text{denominator by} \ 3.\\ & \frac{x^{\color{red}3}}{6^{\color{red}3}} && \text{The answer is in exponential form.}\end{align*}
The answer can be taken one step further. The denominator is numerical so the term can be evaluated.
\begin{align*}& 6^3=6 \times 6 \times 6\\ & 6^3=216\\ & \frac{x^{\color{red}3}}{{\color{red}216}}\\ & \boxed{\left(\frac{x}{6}\right)^3=\frac{x^3}{6^3}=\frac{x^3}{216}}\end{align*}

3.

\begin{align*}& \left(\frac{3x^2}{4y^3}\right)^2 && \text{The base is} \ \frac{3x^2}{4y^3}'.\\ & \frac{3^{1 \times 2} x^{2 \times 2}}{4^{1 \times 2} y^{3 \times 2}} && \text{Keep the base of} \ \frac{3x^2}{4y^3}' \ \text{and multiply the exponents of the factors of both the}\\ & && \text{numerator and the denominator by} \ 2.\\ & \frac{3^{\color{red}4} x^{\color{red}4}}{4^{\color{red}2} y^{\color{red}6}} && \text{The answer is in exponential form.}\end{align*}
The answer can be taken one step further. The denominator and the numerator both have numerical coefficients to be evaluated.
\begin{align*}& 3^2 = 3 \times 3 \qquad \ 4^2=4 \times 4\\ & 3^2=9 \qquad \qquad 4^2=16\\ & \frac{{\color{red}9}x^{\color{red}4}}{{\color{red}16}y^{\color{red}6}}\\ & \boxed{\left(\frac{3x^2}{4y^3}\right)^2=\frac{3^2x^4}{4^2y^6}=\frac{9x^4}{16y^6}}\end{align*}

4.

\begin{align*}& \left(\frac{4a^5b^3}{6ab}\right)^3 && \text{The base is} \ \frac{4a^5b^3}{6ab}'. \ \text{Begin by simplifying the base.}\\ & \frac{2}{3}a^{{\color{red}5-1}}b^{{\color{red}3-1}}=\frac{2a^{\color{red}4}b^{\color{red}2}}{3} && \text{Write the problem with the simplified base.}\\ & \left(\frac{2a^4b^2}{3}\right)^3\\ & \frac{2^{1 \times 3} a^{4 \times 3} b^{2 \times 3}}{3^{1 \times 3}} && \text{Keep the base of} \ \frac{2a^4b^2}{3}' \ \text{and multiply the exponents of the factors of}\\ & && \text{both the numerator and the denominator by} \ 3.\\ & \frac{2^{\color{red}3}a^{{\color{red}12}}b^{\color{red}6}}{3^{\color{red}3}} && \text{The answer is in exponential form.}\end{align*}
The answer can be taken one step further. The denominator and the numerator both have numerical coefficients to be evaluated.
\begin{align*}& 2^3=2 \times 2 \times 2 \quad 3^3 = 3 \times 3 \times 3\\ & 2^3=8 \qquad \qquad \ \ 3^3=27\\ & \frac{{\color{red}8}a^{{\color{red}12}}b^{\color{red}6}}{{\color{red}27}}\\ & \boxed{\left(\frac{4a^5b^3}{6ab}\right)^3=\left(\frac{2a^4b^2}{3}\right)^3=\frac{2^3a^{12}b^6}{3^3}=\frac{8a^{12}b^6}{27}}\end{align*}

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Date Created:
Jan 16, 2013