1. Division Properties of Exponents

2. Warm-Up
Simplify.
(3x4)2
(2a7b)(5a2b4)
2y3 ∙ (4y5)2
9x8
10a9b5
32y13

3. Division Properties of Exponents
Lesson 4.2
Division Properties of Exponents
Use properties of exponents to simplify expressions involving division.

4. Simplify the fractions by cancelling common factors.
Good to Know!
You have learned that when two powers with the same base are multiplied, the base remains the same and the exponents are added together. Examine the division problems below. When two powers with the same base are divided, the base remains the same and the exponents are subtracted. You can also simplify a power of a quotient or fraction by “distributing” the power to both the numerator and denominator.
Simplify the fractions by cancelling common factors.

5. Division Properties of Exponents
Quotient of Powers To divide two powers with the same base, subtract the exponents. Power of a Quotient To find the power of a quotient, find the power of the numerator and denominator.

6. Example 1
Simplify the following. a. Group powers that have the same base. Subtract exponents.

7. Example 1 Continued…
Simplify the following. b. Distribute the power to each base. Multiply powers of powers and evaluate coefficients.

8. Explore! Going Negative
Step 1 Copy the tables below. Find the value of each power with a calculator.
If the value of the power is less than 1, write the power as a fraction.
Step 2 What do you notice about the powers with opposite exponents
(i.e., 22 and 22)?
Step 3 Use your observation from Step 2 to predict the value of each power below.
a. Given that 42 = 16, what is the value of 42?
b. Given that 35 = 243, what is the value of 35?
c. Given that 6 3 = , what is the value of 63?
Power
Value 24 23 22 21
Power
Value
24
23
22
21

9. Explore! Going Negative
Step 4 Look at the statements below. What is the value of each expression (written without an exponent)?
a. b. c.
Step 5 Notice that , and using the Division Property
of Exponents.
Based on your findings in Step 4, what is the value of 50, 20 and 30?
Step 6 Use your calculator to raise other numbers to a power of 0. Try whole numbers and decimal values. What can you conclude?

10. More Properties of Exponents
Negative Exponents For any nonzero number a and integer n, the expression a −n is the reciprocal of an. Also, an is the reciprocal of a −n. Zero Exponents Any nonzero number, a, raised to the zero power is 1. Good to Know! An expression is in simplest form only if it has no negative or zero exponents. An exponent with its base can be moved to the opposite side of the fraction bar to change its sign.

11. Example 2
Simplify the following. a. Any term to the power of 0 equals 1.

12. Example 2 Continued…
Simplify the following. b. Write as separate factors. Use the rules for negative exponents. Multiply factors.

13. Example 2 Continued…
Simplify the following. c. Write as separate factors and group like bases. Simplify by subtracting exponents and dividing coefficients.

14. Example 2 Continued
Simplify the following. c. Use rules for zero and negative exponents. Simplify.

15. Communication Prompt
What do you think is a common error(s) students might make when trying to simplify exponent expressions? How could you help them remember not to make this mistake?

16. Exit Problems
Simplify.