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6.2: Quotient Rules for Exponents

Difficulty Level: Advanced Created by: CK-12
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Practice Exponential Properties Involving Quotients

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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?

Quotient Rules for Exponents

Recall that 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*}

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.

1. 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*}
1. 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*}

Let's simplify the following expressions:

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

\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*}

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

\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*}

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

\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*}

Examples

Example 1

Earlier, you were asked to simplify the following 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*}

This expression 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*}.

Example 2

Simplify the following expression:

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

\begin{align*}\left(\frac{2}{3}\right)^2=\frac{2^2}{3^2}=\frac{4}{9}\end{align*}

Example 3

Simplify the following expression:

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

\begin{align*}\left(\frac{x}{6}\right)^3=\frac{x^3}{6^3}=\frac{x^3}{216}\end{align*}

Example 4

Simplify the following expression:

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

\begin{align*}\left(\frac{3x}{4y}\right)^2=\frac{3^2x^2}{4^2y^2}=\frac{9x^2}{16y^2}\end{align*}

Review

Simplify each of the following expressions, if possible.

1. \begin{align*}\left(\frac{2}{5}\right)^6\end{align*}
2. \begin{align*}\left(\frac{4}{7}\right)^3\end{align*}
3. \begin{align*}\left(\frac{x}{y}\right)^4\end{align*}
4. \begin{align*}\frac{20x^4y^5}{5x^2y^4}\end{align*}
5. \begin{align*}\frac{42x^2y^8z^2}{6xy^4z}\end{align*}
6. \begin{align*}\left(\frac{3x}{4y}\right)^3\end{align*}
7. \begin{align*}\frac{72x^2y^4}{8x^2y^3}\end{align*}
8. \begin{align*}\left(\frac{x}{4}\right)^5\end{align*}
9. \begin{align*}\frac{24x^{14}y^8}{3x^5y^7}\end{align*}
10. \begin{align*}\frac{72x^3y^9}{24xy^6}\end{align*}
11. \begin{align*}\left(\frac{7}{y}\right)^3\end{align*}
12. \begin{align*}\frac{20x^{12}}{-5x^8}\end{align*}
1. Simplify using the laws of exponents: \begin{align*}\frac{2^3}{2^5}\end{align*}
2. Evaluate the numerator and denominator separately and then simplify the fraction: \begin{align*}\frac{2^3}{2^5}\end{align*}
3. Use your result from the previous problem to determine the value of \begin{align*}a\end{align*}: \begin{align*}\frac{2^3}{2^5}=\frac{1}{2^{a}}\end{align*}
4. Use your results from the previous three problems to help you evaluate \begin{align*}2^{-4}\end{align*}.

To see the Review answers, open this PDF file and look for section 6.2.

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Color Highlighted Text Notes

Vocabulary Language: English

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".

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