Math 1B Exponent Rules

Obj: To simplify expressions with zero and negative exponents. Watch Brainpop Video http://www.brainpop.com/math/numbersandoperations/exponents/

Vocabulary PROPERTY: Zero as an Exponent: For every nonzero number a, Negative Exponents: For every nonzero number a and integer n, a-n = 1/an

Demonstrate why #^0 is 1 20 30

Zero and Negative Exponents Simplify. = Use the definition of negative exponent. 1 32 a. 3–2 Simplify. 1 9 = b. (–22.4)0 Use the definition of zero as an exponent. = 1

Zero and Negative Exponents Simplify 1 x –3 a. Rewrite using a division symbol. = 1  x –3 3ab –2 1 b2 Use the definition of negative exponent. = 3a b. = 1  1 x 3 Use the definition of negative exponent. Simplify. 3a b 2 = = 1 • x 3 Multiply by the reciprocal of , which is x 3. 1 x3 = x 3 Identity Property of Multiplication 8-1

Zero and Negative Exponents Evaluate 4x 2y –3 for x = 3 and y = –2. Method 1: Write with positive exponents first. 4x 2y –3 = Use the definition of negative exponent. 4x 2 y 3 Substitute 3 for x and –2 for y. 4(3)2 (–2)3 = 36 –8 –4 1 2 = Simplify. 8-1

Zero and Negative Exponents (continued) Method 2: Substitute first. 4x 2y –3 = 4(3)2(–2)–3 Substitute 3 for x and –2 for y. 4(3)2 (–2)3 = Use the definition of negative exponent. 36 –8 –4 1 2 = Simplify. 8-1

Vocabulary Multiplying Powers with the Same Base: For every nonzero number a and integers m and n, am * an = am+n

Multiplication Properties of Exponents Rewrite each expression using each base only once. Add exponents of powers with the same base. 73 + 2 = a. 73 • 72 = 75 Simplify the sum of the exponents. Think of 4 + 1 – 2 as 4 + 1 + (–2) to add the exponents. 44 + 1 – 2 = b. 44 • 41 • 4–2 = 43 Simplify the sum of the exponents. Add exponents of powers with the same base. 68 + (–8) = c. 68 • 6–8 = 60 Simplify the sum of the exponents. Use the definition of zero as an exponent. = 1 8-3

Multiplication Properties of Exponents Simplify each expression. a. p2 • p • p5 Add exponents of powers with the same base. p 2 + 1 + 5 = = p 8 Simplify. 4x6 • 5x–4 b. Commutative Property of Multiplication (4 • 5)(x 6 • x –4) = Add exponents of powers with the same base. = 20(x 6+(–4)) Simplify. = 20x 2 8-3

Multiplication Properties of Exponents Simplify each expression. a. a 2 • b –4 • a 5 Commutative Property of Multiplication a 2 • a 5 • b –4 = = a 2 + 5 • b –4 Add exponents of powers with the same base. Simplify. a 7 b 4 = Commutative and Associative Properties of Multiplication (2 • 3 • 4)(p 3)(q • q 4) = b. 2q • 3p3 • 4q4 = 24(p 3)(q 1 • q 4) Multiply the coefficients. Write q as q 1. = 24(p 3)(q 1 + 4) Add exponents of powers with the same base. = 24p 3q 5 Simplify. 8-3

Division Properties of Exponents Simplify each expression. x4 x9 = Subtract exponents when dividing powers with the same base. x4 – 9 a. Simplify the exponents. = x–5 Rewrite using positive exponents. 1 x5 = p3 j –4 p–3 j 6 = Subtract exponents when dividing powers with the same base. p3 – (–3)j –4 – 6 b. = p6 j –10 Simplify. Rewrite using positive exponents. p6 j10 = 8-5

Division Properties of Exponents 2 3 –3 a. Simplify . 2 3 –3 = Rewrite using the reciprocal of . 33 23 = Raise the numerator and the denominator to the third power. Simplify. 27 8 3 or = 8-5

Division Properties of Exponents 4b c – b. Simplify . 4b c – –2 2 = Rewrite using the reciprocal of . Write the fraction with a negative numerator. c 4b – 2 = Raise the numerator and denominator to the second power. (–c)2 (4b)2 = Simplify. c2 16b2 =