1. Warm-up: Check your homework!
Lesson 7.3: Define and Use Zero and Negative Exponents Essential Question: How do you evaluate exponential expressions involving zero and negative exponents?
Warm-up: Check your homework!
Common Core
CC.9-12.A.SSE.3c
Use the properties of exponents to transform expressions for exponential functions.

4. Definition of Zero and Negative Exponents
Words
a to the zero power is 1.
𝒂 −𝒏 is the reciprocal of 𝒂 𝒏.
𝒂 𝒏 is the reciprocal of 𝒂 −𝒏 .
Algebra Example

5. Use definition of zero and negative exponents
EXAMPLE 1
Use definition of zero and negative exponents 1 32 =
a. 3– 2
Definition of negative exponents 1 9 =
Evaluate exponent.
b. (–7)0
= 1
Definition of zero exponent

6. Use definition of zero and negative exponents
EXAMPLE 1
Use definition of zero and negative exponents 5 –2 1 c. = 1 5 2
Definition of negative exponents 1 25 =
Evaluate exponent.
= 25
Simplify by multiplying numerator
and denominator by 25. 1
0 5
(Undefined) =
d – 5
a – n is defined only for a nonzero
number a.

7. GUIDED PRACTICE
for Example 1
Evaluate the expression. 2 3 1. 1 2 3. –3 = 1
= 8
2. (–8) – 2 1 64 =
4. (–1 )0
= 1

8. Properties of Exponents
Let a and b be real numbers, and let m and n be integers.
Product of powers property
Power of a power property
Power of a product property
Quotient of powers property
Power of a quotient property

9. Evaluate exponential expressions
EXAMPLE 2
Evaluate exponential expressions
a. 6–
= 6– 4 + 4
Product of a power property
= 60
Add exponents.
= 1
Definition of zero exponent

10. Evaluate exponential expressions
EXAMPLE 2
Evaluate exponential expressions
b. (4– 2)2
= 4– 2 ∙ 2
Power of a power property
= 4– 4
Multiply exponents. 1 4 =
Definition of negative exponents 1
256 =
Evaluate power. c. 1
3– 4
= 34
Definition of negative exponents
= 81
Evaluate power.

11. Evaluate exponential expressions
EXAMPLE 2
Evaluate exponential expressions d.
5– 1 52
= 5– 1 – 2
Quotient of powers property
= 5– 3
Subtract exponents. 1 53 =
Definition of negative exponents 1
125 =
Evaluate power.

12. GUIDED PRACTICE
for Example 2
Evaluate the expression. 5. 1
4– 3
= 64
7. (– 3 )
(– 3 ) – 5 5
= 1 8.
6– 2 62 1
1296 =
6. (5– 3) – 1
= 125

13. Use properties of exponents
EXAMPLE 3
Use properties of exponents
Simplify the expression. Write your answer using only positive exponents.
a. (2xy–5)3
= 23 x3 (y–5)3
Power of a product property
= 8 x3 y–15
Power of a power property =
y15
8x3
Definition of negative exponents

14. Use properties of exponents
EXAMPLE 3
Use properties of exponents
(2x)–2y5
–4x2y2 b. y5
(2x)2(–4x2y2) =
Definition of negative exponents y5
(4x)2(–4x2y2) =
Power of a product property y5
–16x4y2 =
Product of powers property y3
16x4 – =
Quotient of powers property

15. EXAMPLE 4
Standardized Test Practice
The order of magnitude of the mass of a polyphemus moth larva when it hatches is 10-3 gram. During the first 56 days of its life, the moth larva can eat about 105 times its own mass in food. About how many grams of food can the moth larva eat during its first 56 days? A
10–15 gram B
gram C
100 grams D
10,000,000 grams

16. EXAMPLE 4
Standardized Test Practice
SOLUTION
To find the amount of food the moth larva can eat in the first 56 days of its life, multiply its original mass, 10– 3, by 105. –3
= (–3)
= 102
= 100
The moth larva can eat about 100 grams of food in the first 56 days of its life.
ANSWER
The correct answer is C. A B C D

17. GUIDED PRACTICE
for Examples 3 and 4
9x3y
3xy – 3
9. Simplify the expression Write your answer
using only positive exponents. 1
3x2y4
ANSWER
10. SCIENCE The order of magnitude of the mass of a proton is 104 times greater than the order of magnitude of the mass of an electron, which is 10–27 gram. Find the order of magnitude of the mass of a proton.
ANSWER
10 –23 g

18. Classwork/Homework
7.3 Exercises
2-54 even
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