1. Lesson 7-1 Warm-Up

2. “Zero and Negative Exponents” (5-7)
What if zero is an exponent? What if the exponent is a negative number?
Rule: Zero as an Exponent: any nonzero (the base cannot be zero) number, a, with an exponent of zero equals 1. Examples: 50 = 1 (-2)0 = 1 (1.02)0 = 1 (½)0 = 1 a0 = 1 Proof: a4 = a · a · a · a = 1 = 1 a4 = a · a· a · a 1 So a4-4 = a0 = 1 Rule: Negative Exponent: A negative exponent means “the reciprocal of the base number”. In other words, put the base number under 1 to turn a negative exponent into a positive one. Example: Formality: Answers are Written With Only Positive Exponents: To eliminate a negative sign from an exponent, write the reciprocal of the base number. Tip: When evaluating algebraic exponential expressions (algebraic expressions with exponents) for known values, write the expression with positive exponents before substituting known variables with their known values. 1 1 1 1 1 1 1 1 1
a-3 = 1 a3
1 = a3
a-3

3. Use the rule for negative exponents. (–4) = = a. –3 1
Zero and Negative Exponents
LESSON 5-7
Additional Examples
Simplify
(–4) –3 1
(–4) 3
Use the rule for negative exponents.
(–4) = = a. –3 1 1
(-4)(-4)(-4) =
Simplify the denominator. 1
– 64 =
Simplify the denominator.
= – 1 64
Simplify. b.
(–22.4)0
= 1
Use the definition of zero as an exponent.

4. Simplify each expression.
Zero and Negative Exponents
LESSON 7-1
Additional Examples
Simplify each expression. a.
3ab –2
Use the definition of negative exponent. 3a 1 1 b2 =
Simplify. 3a
b 2 = b.
Rewrite using a division symbol.
= 1  x –3 1
x –3
= 1  1
x 3
Use the definition of negative exponent.
= 1 • x 3
Multiply by the reciprocal of , which is x 3. 1 x3
= x 3
Identity Property of Multiplication

5. Evaluate 4x 2y –3 for x = 3 and y = –2.
Zero and Negative Exponents
LESSON 5-7
Additional Examples
Evaluate 4x 2y –3 for x = 3 and y = –2.
Method 1: Write with positive exponents first.
4x 2y –3 =
Write expression in expanded form.
4  x2  y-3 4 1 y2 1 1 x3
Flip negative exponents over to make
them positive exponents. =  
4(3)2
(–2)3 =
Substitute 3 for x and –2 for y.
4(3)(3)
1(-2)(-2)(-2) =
Write exponents in expanded form. 36 –8 =
Simplify.
Make improper fractions into mixed
Numbers by dividing bottom into top. 1 2 = –4

6. Evaluate 4x 2y –3 for x = 3 and y = –2.
Zero and Negative Exponents
LESSON 5-7
Additional Examples
Evaluate 4x 2y –3 for x = 3 and y = –2.
Method 2: Substitute first.
4x 2y –3 =
Write expression in expanded form and
substitute 3 for x and –2 for y. .
4  (3)2  (-2)-3 4 1
(3)2 1 1
(-2)3
Flip negative exponents over to make
them positive exponents. =  
4(3)(3)
1(-2)(-2)(-2) =
Write exponents in expanded form. 36 –8 =
Simplify.
Make improper fractions into mixed
Numbers by dividing bottom into top. 1 2 = –4

7. a. Evaluate the expression for m = 0.
Zero and Negative Exponents
LESSON 7-1
Additional Examples
In the lab, the population of a certain bacteria doubles every month. The expression 3000 • 2m models a population of 3000 bacteria after m months of growth. Evaluate the expression for m = 0 and m = –2. Describe what the value of the expression represents in each situation.
a. Evaluate the expression for m = 0.
3000 • 2m = 3000 • 20   Substitute 0 for m.
= 3000 • 1 Simplify.
= 3000
When m = 0, the value of the expression is This represents the initial population of the bacteria. This makes sense because when m = 0, no time has passed.

8. b. Evaluate the expression for m = –2.
Zero and Negative Exponents
LESSON 7-1
Additional Examples
(continued)
b. Evaluate the expression for m = –2.
3000 • 2m = 3000 • 2–2 Substitute –2 for m.
= 3000 • Simplify. 1 22
3000 1 1 4 = • =
750
When m = –2, the value of the expression is 750. This represents the 750 bacteria in the population 2 months before the present population of 3000 bacteria.

9. Simplify each expression. 1. 3–4 2. (–6)0
Zero and Negative Exponents
LESSON 7-1
Lesson Quiz
Simplify each expression.
1. 3–4 2. (–6)0
3. –2a0b–2 4.
• • 3–2
1 81 1 2 b2 – k
m–3
km3
8000
500