1. Copyright © Cengage Learning. All rights reserved. Polynomials 4

2. Copyright © Cengage Learning. All rights reserved. Section 4.1 Natural-Number Exponents

3. 3 Objectives Write an exponential expression without exponents. Write a repeated multiplication expression using exponents. Simplify an expression by using the product rule for exponents. Simplify an expression by using the power rules for exponents. 1 1 2 2 3 3 4 4

4. 4 Objectives 1. Simplify an expression by using the quotient 2. rules for exponents. 5 5

5. 5 Write an exponential expression without exponents 1.

6. 6 Japanese Sword Making Japanese sword making Begins with raw iron (1:15-3:07) Forged and folded 15 times. (4:34-8:06) How many layers does this produce?

7. 7 Write an exponential expression without exponents We have used natural-number exponents to indicate repeated multiplication. For example, 2 5 = 2  2  2  2  2 = 32 (–7) 3 = (–7)(–7)(–7) = –343 x 4 = x  x  x  x –y 5 = –y  y  y  y  y These examples suggest a definition for x n, where n is a natural number.

8. 8 Write an exponential expression without exponents Natural-Number Exponents If n is a natural number, then x n = x  x  x  …  x In the exponential expression x n, x is called the base and n is called the exponent. The entire expression is called a power of x. x n n factors of x Base Exponent

9. 9 Write an exponential expression without exponents If an exponent is a natural number, it tells how many times its base is to be used as a factor. An exponent of 1 indicates that its base is to be used one time as a factor, an exponent of 2 indicates that its base is to be used two times as a factor, and so on. 3 1 = 3 (–y) 1 = –y (–4z) 2 = (–4z)(–4z) (t 2 ) 3 = t 2  t 2  t 2

10. 10 Example Find each value to show that a. (–2) 4 and b. –2 4 have different values. Solution: We find each power and show that the results are different. a. (–2) 4 = (–2)(–2)(–2)(–2) = 16

11. 11 Example 1 – Solution b. –2 4 = –(2 4 ) = –(2  2  2  2) = –16 Since 16  –16, it follows that –(2) 4  –2 4. Note that in part a the base is –2 but in part b the base is 2. cont’d

12. 12 Write an exponential expression without exponents Comment There is a pattern regarding even and odd exponents. If the exponent of a base is even, the result is positive. If the exponent of a base is odd, the result will be the same sign as the original base.

13. 13 Write a repeated multiplication expression using exponents 2.

14. 14 Example Write each expression using one exponent. a. 3  3  3  3  3 b. (5z)(5z)(5z) Solution: a. Since 3 is used as a factor five times, 3  3  3  3  3 = 3 5 b. Since 5z is used as a factor three times, (5z)(5z)(5z) = (5z) 3

15. 15 Simplify an expression by using the product rule for exponents 3.

16. 16 Simplify an expression by using the product rule for exponents To develop a pattern for multiplying exponential expressions with the same base, we consider the product x 2  x 3. Since the expression x 2 means that x is to be used as a factor two times and the expression x 3 means that x is to be used as a factor three times, we have x 2 x 3 = x  x  x  x  x = x  x  x  x  x = x 5 2 factors of x 3 factors of x 5 factors of x

17. 17 Simplify an expression by using the product rule for exponents In general, x m  x n = x  x  x ...  x  x  x  x ...  x = x  x  x  x  x  x ...  x  x  x = x m + n m factors of x n factors of x m + n factors of x

18. 18 Simplify an expression by using the product rule for exponents This discussion suggests the following pattern: To multiply two exponential expressions with the same base, keep the base and add the exponents. Product Rule for Exponents If m and n are natural numbers, then x m x n = x m + n

19. 19 Example Simplify each expression. a. x 3 x 4 = x 3 + 4 = x 7 b. y 2 y 4 y = (y 2 y 4 )y = (y 2 + 4 )y = y 6 y = y 6 + 1 = y 7 Keep the base and add the exponents. 3 + 4 = 7 Use the associative property to group y 2 and y 4 together. Keep the base and add the exponents. 2 + 4 = 6 Keep the base and add the exponents: y = y 1. 6 + 1 = 7

20. 20 Simplify an expression by using the product rule for exponents Comment The product rule for exponents applies only to exponential expressions with the same base. An expression such as x 2 y 3 cannot be simplified, because x 2 and y 3 have different bases.

21. 21 Simplify an expression by using the power rules for exponents 4.

22. 22 Simplify an expression by using the power rules for exponents To find another pattern of exponents, we consider the expression (x 3 ) 4, which can be written as x 3  x 3  x 3  x 3. Because each of the four factors of x 3 contains three factors of x, there are 4  3 (or 12) factors of x. Thus, the expression can be written as x 12. (x 3 ) 4 = x 3  x 3  x 3  x 3 = x  x  x  x  x  x  x  x  x  x  x  x = x 12 12 factors of x x3x3 x3x3 x3x3 x3x3

23. 23 Simplify an expression by using the power rules for exponents In general, (x m ) n = x m  x m  x m ...  x m = x  x  x  x  x  x  x ...  x = x m  n n factors of x m m  n factors of x

24. 24 Simplify an expression by using the power rules for exponents The previous discussion suggests the following pattern: To raise an exponential expression to a power, keep the base and multiply the exponents. Power Rule for Exponents If m and n are natural numbers, then (x m ) n = x mn

25. 25 Example Simplify: a. (2 3 ) 7 = 2 3  7 = 2 21 b. (z 7 ) 7 = z 7  7 = z 49 Keep the base and multiply the exponents. 3  7 = 21 Keep the base and multiply the exponents. 7  7 = 49

26. 26 Simplify an expression by using the power rules for exponents To find more patterns for exponents, we consider the expressions (2x) 3 and. (2x) 3 = (2x)(2x)(2x) = (2  2  2)(x  x  x) = 2 3 x 3 = 8x 3

27. 27 Simplify an expression by using the power rules for exponents

28. 28 Simplify an expression by using the power rules for exponents These examples suggest the following patterns: To raise a product to a power, we raise each factor of the product to that power, and to raise a quotient to a power, we raise both the numerator and denominator to that power. Product to a Power Rule for Exponents If n is a natural number, then (xy) n = x n y n Quotient to a Power Rule for Exponents If n is a natural number, and if y  0, then

29. 29 Simplify an expression by using the quotient rule for exponents 5.

30. 30 Simplify an expression by using the quotient rule for exponents To find a pattern for dividing exponential expressions, we consider the fraction where the exponent in the numerator is greater than the exponent in the denominator. We can simplify the fraction as follows:

31. 31 Simplify an expression by using the quotient rule for exponents The result of 4 3 has a base of 4 and an exponent of 5 – 2 (or 3). This suggests that to divide exponential expressions with the same base, we keep the base and subtract the exponents. Quotient Rule for Exponents If m and n are natural numbers, m  n and x  0, then

32. 32 Example Simplify. Assume no division by zero.

33. 33 Example cont’d