Chapter 5 Exponents and Polynomials 5 1 Exponents

Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3

Evaluating Exponential Expressions Example: Evaluate each of the following expressions. 34 = 3 •

Evaluating Exponential Expressions Example: Evaluate each of the following expressions. a. ) Find 3

The Product Rule for Exponents If m and n are positive integers and a

The Power Rule for Exponents If m and n are positive integers and a

The Power of a Product Rule If n is a positive integer and a

The Power of a Quotient Rule If n is a positive integer and a

The Quotient Rule for Exponents If m and n are positive integers and a

Zero Exponent a 0 = 1, as long as a is not 0. Example:

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Chapter 5 Exponents and Polynomials

§ 5. 1 Exponents

Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 • 3 • 3 3 is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 3

Evaluating Exponential Expressions Example: Evaluate each of the following expressions. 34 = 3 • 3 • 3 = 81 (– 5)2 = (– 5) = 25 – 62 = – (6)(6) = – 36 (2 • 4)3 = (2 • 4)(2 • 4) = 8 • 8 = 512 3 • 42 = 3 • 4 = 48 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 4

Evaluating Exponential Expressions Example: Evaluate each of the following expressions. a. ) Find 3 x 2 when x = 5. 3 x 2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75 b. ) Find – 2 x 2 when x = – 1. – 2 x 2 = – 2(– 1)(– 1) = – 2(1) = – 2 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 5

The Product Rule for Exponents If m and n are positive integers and a is a real number, then am · an = am+n Example: Simplify each of the following expressions. 32 · 34 = 32+4 = 36 = 3 · 3 · 3 · 3= 729 x 4 · x 5 = x 4+5 = x 9 z 3 · z 2 · z 5= z 3+2+5 = z 10 (3 y 2)(– 4 y 4) = 3 · y 2 (– 4) · y 4 = 3(– 4)(y 2 · y 4) = – 12 y 6 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 6

The Power Rule for Exponents If m and n are positive integers and a is a real number, then (am)n = amn Example: Simplify each of the following expressions. (23)3 = 23· 3 = 29 = 512 (x 4)2 = x 4· 2 = x 8 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 7

The Power of a Product Rule If n is a positive integer and a and b are real numbers, then (ab)n = an · bn Example: Simplify (5 x 2 y)3 = 53 · (x 2)3 · y 3 = 125 x 6 y 3 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 8

The Power of a Quotient Rule If n is a positive integer and a and c are real numbers, then Example: Simplify Martin-Gay, Beginning and Intermediate Algebra, 4 ed 9

The Quotient Rule for Exponents If m and n are positive integers and a is a real number, then as long as a is not 0. Example: Simplify the following expression. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 10

Zero Exponent a 0 = 1, as long as a is not 0. Example: Simplify each of the following expressions. 50 = 1 (xyz 3)0 = x 0 · y 0 · (z 3)0 = 1 · 1 = 1 –x 0 = –(x 0) = – 1 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 11