Lesson 7-1 Warm-Up

“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

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.

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

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

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

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 3000. This represents the initial population of the bacteria. This makes sense because when m = 0, no time has passed.

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.

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. 5. 8000 • 40 6. 4500 • 3–2 1 81 1 2 b2 – k m–3 km3 8000 500