# 8.2: Exponent Properties Involving Quotients

## 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.*

Solution:

Notice also that:

by the quotient rule.

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.*

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

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

Additional Examples:

a.

b.

c.

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.

b.

c.

## 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.