# 8.2: Exponent Properties Involving Quotients

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Use the quotient of powers property.
- Use the power of a quotient property.
- Simplify expressions involving quotient properties of exponents.

## Vocabulary

Terms introduced in this lesson:

- quotient rule for exponents
- power rule for quotients

## Teaching Strategies and Tips

Allow students to infer in Example 1 that *canceling* like factors is equivalent to finding the difference in the exponents of the factors.

- Therefore, the quotient rule for exponents is based on
*canceling*like factors.

Additional Example:

*Simplify the expression using the quotient rule.*

\begin{align*}\frac{x^4} {x^3}.\end{align*}

Solution:

\begin{align*}\frac{x^4} {x^3} = \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x} {\cancel{x} \cdot \cancel{x} \cdot \cancel{x}} = \frac{x} {1} = x.\end{align*}

Notice also that:

\begin{align*}\frac{x^4} {x^3} = x^{4 - 3} = x^1 = x\end{align*}

Use Example 2 to show what happens when a larger exponent is subtracted from a smaller exponent.

- Encourage students to do the division longhand.
- Allow students to come to the conclusion that positive exponents result when subtracting smaller exponents from larger exponents.

Additional Example:

*Simplify the expression leaving all powers positive.*

\begin{align*}\frac{m^3n^2} {m^4n^3}.\end{align*}

Solution: Subtract the smaller exponents from the larger exponents and leave the result in the denominator (where the larger exponents were).

\begin{align*}\frac{m^3n^2} {m^4n^3} = \frac{1} {m^{4 - 3} n^{3 - 2}} = \frac{1} {m^1n^1} = \frac{1} {mn}.\end{align*}

Use Examples 1c, 3b, and 6 to show that the quotient rule is applied separately for each factor in the expression.

Additional Examples:

a. \begin{align*}\frac{2xy^2} {6x^3y}\end{align*}

b. \begin{align*}\frac{72x^5 y^2 z} {144x^3 y^2 z^4}\end{align*}

c. \begin{align*}\frac{56x^2 y^2 z^5} {18x^8 yz}\end{align*}

Combine the quotient and power rules in Examples 5-7.

- Suggest that students include each step in their solution.
- Remind students to follow the order of operations.

Additional Examples:

a. \begin{align*}\left (\frac{24m^3} {10m^4}\right )^3 \cdot \left ( \frac{n^{12}} {n^5}\right )^2\end{align*}

b. \begin{align*}\left (\frac{12x^2} {18y^3}\right )^2 \cdot \frac{x^3} {x^2}\end{align*}

c. \begin{align*}\left (\frac{65x^5 y^2} {145x^3 y^2}\right )^3\end{align*}

## Error Troubleshooting

In *Review Question* 7, have students apply the quotient rule first. This will save students some work in simplifying because the numbers will be smaller. See also Examples 4a and 4b; and *Review Questions* 11, 13, and 14.

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