Properties of Exponents

1. Section 4.1 Properties of Exponents

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

3. Definition Two powers of b have specific names. We refer to b2 as the square of b or b squared. We refer to b3 as the cube of bor b cubed. For –bn, we compute bn before finding the opposite. For –24, the base is 2, not –2. If we want the base –2 Clarify Section 4.1 Slide 3 Definition: Exponent Definition of an Exponent

4. Calculator Use a graphing calculator to check both computations To find –24, press (–) 2 ^ 3 ENTER Section 4.1 Slide 4 Definition: Exponent Definition of an Exponent

5. Properties Section 4.1 Slide 5 Properties of Exponents Properties of Exponent

6. Example Show that b5b3 = b5. Writing b5b3 without exponents, we see that Use calculator to verify by using various bases and examining the table Solution Section 4.1 Slide 6 Properties of Exponents Properties of Exponent

7. Solution Continued Show that bmbn = bm+n, where m and n are counting numbers. Example Section 4.1 Slide 7 Properties of Exponents Properties of Exponent

8. Solution Write bmbnwithout exponents: Show that , n is a counting number and c ≠ 0. Example Section 4.1 Slide 8 Properties of Exponents Properties of Exponent

9. Solution Write without exponents: Section 4.1 Slide 9 Properties of Exponents Properties of Exponent

10. Property An expression involving exponents is simplified if It includes no parentheses. Each variable or constant appears as a base as few times as possible. For example, we write x2x4 = x6 Each numerical expression (such as 72) has been calculated, and each numerical fraction has been simplified. Each exponent is positive. Section 4.1 Slide 10 Simplifying Expressions Involving Exponents Simplifying Expressions Involving Exponents

11. Example Simplify. Section 4.1 Slide 11 Simplifying Expressions Involving Exponents Simplifying Expressions Involving Exponents

12. Solution Section 4.1 Slide 12 Simplifying Expressions Involving Exponents Simplifying Expressions Involving Exponents

13. Solution Continued Section 4.1 Slide 13 Simplifying Expressions Involving Exponents Simplifying Expressions Involving Exponents

14. Warning 3b2 and (3b)2 are not equivalent 3b2 base is b, and (3b)2 base is the 3b Typical error looks like Section 4.1 Slide 14 Simplifying Expressions Involving Exponents Simplifying Expressions Involving Exponents

15. Introduction What is the meaning of b0? The property is to be true for m = n, then So, a reasonable definition of b0 is 1. For b≠ 0, b0= 1 Definition Section 4.1 Slide 15 Simplifying Expressions Involving Exponents Zero as an Exponent

16. Illustration 70 = 1, (–3)0 = 1, and (ab)0 = 1, where ab ≠ 0 Section 4.1 Slide 16 Simplifying Expressions Involving Exponents Zero as an Exponent

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

18. Definition 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 Illustration Section 4.1 Slide 18 Negative Integer Exponents Negative Exponents

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

20. Definition 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, Example Section 4.1 Slide 20 Negative Exponents Negative Exponents

21. Example Simplify. Solution Section 4.1 Slide 21 Simplifying More Expressions Involving Exponents Simplify More Expressions Involving Exponents

22. Properties Section 4.1 Slide 22 Properties of Integer Exponents Simplify More Expressions Involving Exponents