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**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.

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**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

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**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

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**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.

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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

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**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

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**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

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**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.

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**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.

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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

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**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

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**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

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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

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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

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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

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Classwork/Homework 7.3 Exercises 2-54 even Pages

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