LESSON 6.1 EXPONENTS Overview Rosa plans to invest $1000 in an Individual Retirement Account (IRA). She can invest in bonds that offer a return of 7% annually, or a riskier stock fund that is expected to return 10% annually. Rosa would like to know how much her money can grow in 30 years. Exponents can help her answer this question. In this lesson, you will study exponents and their properties. Explain Concept 1 has sections on CONCEPT 1: • Exponential Notation PROPERTIES OF EXPONENTS •Multiplication Property • Division Property Exponential Notation Exponents are used to indicate repeated multiplication of the same • Power of a Power Property number. • Power of a Product For example, we use exponential notation to write: Property 5 и 5 и 5 и 5 ϭ 54 • Power of a Quotient Property 54 is read “five to the fourth power.” • Zero Power Property In the expression 54: •Using Several Properties of • The base, 5, is the repeated factor. Exponents • The exponent, 4, indicates the number of times the base appears as a factor. An exponent is also called a power. Exponent Base 54 ϭ 5 и 5 и 5 и 5 ϭ 625 ͕ 4 factors Product LESSON 6.1 EXPONENTS EXPLAIN 367 Example 6.1.1 Find: 23 Solution The base is 2 and the exponent is 3. 23 = 2 и 2 и 2 = 8 ͕ 3 factors Example 6.1.2 Rewrite using exponential notation: 10 и 10 и 10 и 10 и 10 и 10 Solution 6 There are six factors. Each is 10. 10 и 10 и 10 10 и 10 и 10 = 10 Therefore, the base is 10 and the ͕ 6 factors exponent is 6. Exponents have several properties. We will use these properties to simplify expressions. In the properties that follow, each variable represents a real number. Multiplication Property — Property — Multiplication Property of Exponents English To multiply two exponential expressions with the same base, add their exponents. The base stays the same. Algebra xm и xn ϭ xm ϩ n (Here, m and n are positive integers.) Example 54 и 52 ϭ 54 ϩ 2 ϭ 5 6 Example 6.1.3 a. Use the Multiplication Property of Exponents to simplify 23 и 24. b. Use the definition of exponential notation to justify your answer. Solution Remember to add the exponents, a. The operation is multiplication and the bases are the same. but leave the bases alone. Therefore, add the exponents and use 2 as the base. That is, 23 и 24 ϭ 23 ϩ 4 ϭ 27, not 47. 23 и 24 ϭ 23 ϩ 4 ϭ 27 Note the difference between b. Rewrite the product to show the factors. Then simplify. 23 и 24 and 23 ϩ 24. 3 4 7 2 и 2 = (2 и 2 и 2) (2 и 2 и 2 и 2) = 2 и 2 и 2 и 2 и 2 и 2 и 2 = 2 ϩ ͕ ͕ 23 и 24 ϭ 23 4 ϭ 27 ϭ 128 ͕ 23 ϩ 24 ϭ 8 ϩ 16 ϭ 24 3 factors 4 factors 7 factors 368 TOPIC 6 EXPONENTS AND POLYNOMIALS — Caution — Negative Bases A negative sign is part of the base only when the negative sign is inside the parentheses that enclose the base. For example, consider the following cases: .In (؊3)2, the base is ؊3. In Ϫ32, the base is 3 ؊3)2 ϭ (؊3) и (؊3) ϭϩ9You can think of Ϫ32 as the) “opposite” of 32. Ϫ32 ϭϪ(3 и 3) ϭϪ9 Example 6.1.4 If possible, use the Multiplication Property of Exponents to simplify each expression: a. (Ϫ2)2 и (Ϫ2)4 b. Ϫ22 и 24 c. Ϫ22 и (Ϫ2)4 Solution a. In (Ϫ2)2 и (Ϫ2)4, the base (Ϫ2)2 и (Ϫ2)4 ϭ (Ϫ2)2 ϩ 4 is Ϫ2. ϭ (Ϫ2)6 ϭ 64 b. In Ϫ22 и 24, the base is 2. Ϫ22 и 24 ϭϪ(22) и (24) We may think of Ϫ22 и 24 ϭϪ(22 ϩ 4) as the opposite of 22 и 24. ϭϪ(26) ϭϪ64 c. In Ϫ22 и (Ϫ2)4, the base of the first factor, Ϫ22, is 2. The base of the second factor, (Ϫ2)4, is Ϫ2. The bases are not the same, so we cannot use the Multiplication Property of Exponents. However, we can still evaluate the expression. Ϫ22 и (Ϫ2)4 ϭϪ4 и 16 ϭϪ64 We can extend the Multiplication Property of Exponents to multiply more than two factors. Example 6.1.5 Find: 84 и 8 и 85. Leave your answer in exponential notation. We left 810 in exponential form. To Solution evaluate 810, use the “yx ” key on a The bases are the same, so we can use the scientific calculator or the “^” key Multiplication Property of Exponents. on a graphing calculator. 10 ϭ Note: 8 ϭ 81 84 и 8 и 85 ϭ 84 и 81 и 85 ϭ 84 ϩ 1 ϩ 5 ϭ 810 8 1,073,741,824 LESSON 6.1 EXPONENTS EXPLAIN 369 Example 6.1.6 Find: x7 и x3 и x5 Solution The operation is multiplication and the bases are the same. Therefore, add the exponents and x7 и x3 и x5 ϭ x7 ϩ 3 ϩ 5 ϭ x15 use x as the base. Division Property — Property — Division Property of Exponents English To divide two exponential expressions with the same base: Compare the exponents. • If the greater exponent is in the numerator, write the base in the numerator. • If the greater base is in the denominator, write the base in the denominator. Then subtract the smaller exponent from the greater. Use the result as the new exponent. Algebra Example xm 25 ᎏᎏ ϭ xm Ϫ n for m Ͼ n and x 0. ᎏᎏ ϭ 25 Ϫ 3 ϭ 22 xn 23 xm 1 23 1 1 ᎏᎏ ϭ ᎏᎏ for m Ͻ n and x 0. ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ xn xn Ϫ m 25 25 Ϫ 3 22 (Here, m and n are positive integers.) Example 6.1.7 xm 1 53 Since 3 Ͻ 4, we use the form ᎏᎏ ϭ ᎏᎏ . a. Use the Division Property of Exponents to find ᎏᎏ. xn xn Ϫ m 54 b. Use the definition of exponential notation to justify your answer. Solution 53 1 1 a. The bases are the same, ᎏᎏ¬ϭ ᎏᎏ ϭ ᎏᎏ 54 54 Ϫ 3 5 so subtract the exponents. 53 5 и 5 и 5 b. Rewrite the numerator and ᎏᎏ¬ϭ ᎏᎏ 54 5 и 5 и 5 и 5 denominator to show the factors. 1 1 1 5΋ и 5΋ и 5΋ 1 Cancel the common factors. ¬ϭ ᎏᎏ ϭ ᎏᎏ 5΋ и 5΋ и 5΋ и 5 5 1 1 1 370 TOPIC 6 EXPONENTS AND POLYNOMIALS Example 6.1.8 Find: 79 Ϭ 76. Leave your answer in exponential notation. Solution The operation is division and the 9 xm 9 6 7 9 Ϫ 6 3 Ͼ ᎏᎏ ϭ m Ϫ n bases are the same. Therefore, subtract 7 Ϭ 7 ϭ ᎏᎏ ϭ 7 ϭ 7 Since 9 6, we use the form n x . 76 x the exponents and use 7 as the base. Example 6.1.9 Find: w8 Ϭ w13 Solution The operation is division and the 8 m 8 13 w 1 1 Ͻ ᎏxᎏ ϭ ᎏ1ᎏ bases are the same. Therefore, subtract w Ϭ w ϭ ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ Since 8 13, we use the form n n Ϫ m . w13 w13 Ϫ 8 w5 x x the exponents and use w as the base. Power of a Power Property — Property — Power of a Power Property of Exponents English To raise a power to a power, multiply the exponents. Algebra (xm)n ϭ xmn (Here, m and n are positive integers.) Example (72)4 ϭ 72 и 4 ϭ 78 Example 6.1.10 a. Use the Power of a Power Property of Exponents to simplify (52)3. b. Use the definition of exponential notation to justify your answer. Solution a. To raise a power to a power, (52)3 ϭ 52 и 3 ϭ 56 multiply the exponents. b. Rewrite each power to show the factors. Then simplify. 3 ( 52) = ( 52) ( 52) ( 52) = (5 и 5) и (5 и 5) и (5 и 5) = 56 ͕ ͕ 3 factors 6 factors LESSON 6.1 EXPONENTS EXPLAIN 371 Example 6.1.11 Simplify: (y5)3 Solution To simplify a power of a power, (y5)3 ϭ y5 и 3 ϭ y15 multiply the exponents. Power of a Product Property — Property — Power of a Product Property of Exponents English To raise a product to a power, you can first raise each factor to the power. Then multiply. Algebra (xy)n ϭ xnyn (Here, n is a positive integer.) Example (2x)3 ϭ 23x3 ϭ 8x3 Example 6.1.12 a. Use the Power of a Product Property of Exponents to simplify (3y)2. b. Use the definition of exponential notation to justify your answer. Solution a. Raise each factor to the power 2. (3y)2¬ϭ 32y2 ϭ 9y2 b. Rewrite the power to show (3y)2¬ϭ (3y) и (3y) the factors. Then simplify. ¬ϭ 3 и 3 и y и y ¬ϭ 32y2 ¬ϭ 9y2 Example 6.1.13 Simplify: (23 и w5)4 Solution Use the Power of a Product Property of Exponents to raise each factor inside (23 и w5)4¬ϭ (23)4(w5)4 the parentheses to the power 4. Use the Power of a Power Property ¬ϭ (23 и 4)(w5 и 4) of Exponents.