Properties of Exponents Section 4.1

Lehmann, Intermediate Algebra, 4ed Section 4.1 For any counting number n, We refer to b n at the power; the nth power of b, or b raised to the nth power. We call b the base and n the exponent. Slide 2 Definition: Exponent Definition of an Exponent Definition N factors of b

Lehmann, Intermediate Algebra, 4ed Section 4.1 Two powers of b have specific names. We refer to b 2 as the square of b or b squared. We refer to b 3 as the cube of b or b cubed. For –b n, we compute b n before finding the opposite. For –2 4, the base is 2, not –2. If we want the base –2 Slide 3 Definition: Exponent Definition of an Exponent Definition Clarify

Lehmann, Intermediate Algebra, 4ed Section 4.1 Use a graphing calculator to check both computations To find –2 4, press (–) 2 ^ 3 ENTER Slide 4 Definition: Exponent Definition of an Exponent Calculator

Lehmann, Intermediate Algebra, 4ed Section 4.1Slide 5 Properties of Exponents Properties of Exponent Properties

Lehmann, Intermediate Algebra, 4ed Section 4.1 Show that b 5 b 3 = b 5. Writing b 5 b 3 without exponents, we see that Use calculator to verify by using various bases and examining the table Slide 6 Properties of Exponents Properties of Exponent Example Solution

Lehmann, Intermediate Algebra, 4ed Section 4.1 Show that b m b n = b m+n, where m and n are counting numbers. Slide 7 Properties of Exponents Properties of Exponent Example Solution Continued

Lehmann, Intermediate Algebra, 4ed Section 4.1 Write b m b n without exponents: Show that, n is a counting number and c ≠ 0. Slide 8 Properties of Exponents Properties of Exponent Solution Example

Lehmann, Intermediate Algebra, 4ed Section 4.1 Write without exponents: Slide 9 Properties of Exponents Properties of Exponent Solution

Lehmann, Intermediate Algebra, 4ed Section 4.1 An expression involving exponents is simplified if 1.It includes no parentheses. 2.Each variable or constant appears as a base as few times as possible. For example, we write x 2 x 4 = x 6 3.Each numerical expression (such as 7 2 ) has been calculated, and each numerical fraction has been simplified. 4.Each exponent is positive. Slide 10 Simplifying Expressions Involving Exponents Property

Lehmann, Intermediate Algebra, 4ed Section 4.1 Simplify. Slide 11 Simplifying Expressions Involving Exponents Example

Lehmann, Intermediate Algebra, 4ed Section 4.1Slide 12 Simplifying Expressions Involving Exponents Solution

Lehmann, Intermediate Algebra, 4ed Section 4.1Slide 13 Simplifying Expressions Involving Exponents Solution Continued

Lehmann, Intermediate Algebra, 4ed Section 4.1 3b 2 and (3b) 2 are not equivalent 3b 2 base is b, and (3b) 2 base is the 3b Typical error looks like Slide 14 Simplifying Expressions Involving Exponents Warning

Lehmann, Intermediate Algebra, 4ed Section 4.1 What is the meaning of b 0 ? The property is to be true for m = n, then So, a reasonable definition of b 0 is 1. For b ≠ 0, b 0 = 1 Slide 15 Simplifying Expressions Involving Exponents Zero as an Exponent Introduction Definition

Lehmann, Intermediate Algebra, 4ed Section = 1, (–3) 0 = 1, and (ab) 0 = 1, where ab ≠ 0 Slide 16 Simplifying Expressions Involving Exponents Zero as an Exponent Illustration

Lehmann, Intermediate Algebra, 4ed Section 4.1 If n is a negative integer, what is the meaning of b n ? What is the meaning of a negative exponent? If the property is true for m = 0, then So, we would define b –n to be. Slide 17 Negative Exponents Introduction

Lehmann, Intermediate Algebra, 4ed Section 4.1 If b ≠ 0 and n is a counting number, then In words: To find b –n, take its reciprocal and switch the sign of the exponent. For example Slide 18 Negative Integer Exponents Negative Exponents Definition Illustration

Lehmann, Intermediate Algebra, 4ed Section 4.1 We write in another form, where b ≠ 0 and n is a counting number: Slide 19 Negative Exponents Introduction

Lehmann, Intermediate Algebra, 4ed Section 4.1 If b ≠ 0 and n is a counting number, then In words: To find, take its reciprocal and switch t he sign of the exponent. For example, Slide 20 Negative Exponents Definition Example

Lehmann, Intermediate Algebra, 4ed Section 4.1 Simplify. Slide 21 Simplifying More Expressions Involving Exponents Simplify More Expressions Involving Exponents Example Solution

Lehmann, Intermediate Algebra, 4ed Section 4.1Slide 22 Properties of Integer Exponents Simplify More Expressions Involving Exponents Properties