#### Transcript Lesson 6.5: Looking Back with Exponents

• To review or learn the division property of exponents • In the previous lesson you learned that looking ahead in time to predict future growth with an exponential model is related to the multiplication property of exponents. • In this lesson you’ll discover a rule for dividing expressions with exponents. Then you’ll see how dividing expressions is like looking back in time. The Division Property of Exponents • Step 1: ▫ Write the numerator and the denominator of each quotient in expanded form. ▫ Then reduce to eliminate common factors. ▫ Rewrite the factors that remain with exponents. Use your calculator to check your answers. 9 5 56 3 3 3 5 3 52 44 x 6 42 x 3 9 5 56 1 1 1 1 1 1 1 1 5555555555 5 5 5 5 5 5 555 53 555555 5 5 5 5 5 5 1 1 1 1 1 1 1 33 53 3 3 3 5 5 5 3 3 3 5 5 5 32 5 2 3 5 2 35 355 1 3 5 5 1 44 x 6 42 x 3 1 1 4444 x x x x x x 44 x x x 1 1 1 1 1 4 4 4 4 x x x x x x 4 4 x x x 1 1 42 x 3 42 x 3 1 1 1 1 • Step 2: ▫ Compare the exponents in each final expression you got in Step 1 to the exponents in the original quotient. ▫ Describe a way to find the exponents in the final expression without using expanded form. 59 3 5 56 33 53 2 3 5 2 35 44 x 6 2 3 4 x 2 3 4 x • Step 3: ▫ Use your method from Step 2 to rewrite this expression so that it is not a fraction. 0.08 You can leave 12 as a fraction. 515 1 511 1 24 0.08 12 18 0.08 12 5 1 11 5 1 15 24 0.08 12 18 0.08 12 24 18 15 11 5 0.08 1 12 0.08 5 1 12 4 6 • Recall that exponential growth is related to repeated multiplication. When you look ahead in time you multiply by repeated constant multiplication, or increase the exponent. • To look back in time you will need to undo some of the constant multipliers, or divide. • Step 4 ▫ Apply what you have discovered about dividing expressions with exponents. After 7 years the balance in a saving account is 500(1+0.04)7. What does the expression 500 1 0.04 mean in this situation? 3 1 0.04 7 Rewrite this expression with a single exponent. The balance 3 years prior. 500 1 0.04 4 ▫ After 9 years of depreciation, the value of a car is 9 21,300 1-0.12 , 1-0.12 5 21,300 1-0.12 9 ▫ What does the expression mean in this 5 1-0.12 situation? ▫ Rewrite this expression with a single exponent. The balance 5 years prior 21,300 1-0.12 4 ▫ After 5 weeks the population of a bug colony is 32(1+0.50)5. ▫ Write a division expression to show the population 2 weeks earlier. ▫ Rewrite your expression with a single exponent. 32 1 0.50 5 1 0.50 2 32 1 0.50 3 ▫ The expression A(1+r)n can model n time periods of exponential growth. What expression models the growth m time periods earlier. A 1 r n 1 r m A 1 r n-m • Step 5 ▫ How does looking back in time with an exponential model relate to dividing expressions with exponents? Dividing by (1+r)m represents looking back m time periods. Expanded form helps you understand many properties of exponents. It also helps you understand how the properties work together. Example A • Use the properties of exponents to rewrite each expression. 6 x 9 6 x x x x x x x x x 6 x x x x x 6 x5 4 5 x x x x 5 5 5x 3x 8x 2 4 4 x 3 7.5 x108 1.5 x103 3 8 x2 x 4 3 8 x x x x x x 3 6 x 4 xxx 4 x 3 7.5 x108 7.5 10 10 10 10 10 10 10 10 5 5.0 x10 10 10 10 1.5 x103 1.5 For any non-zero value of b and any integer value of m and n n b nm b m b Example B • Six years ago, Anne bought a van for $18,500 for her flower delivery service. Based on the prices of similar used vans, she estimates a rate of depreciation of 9% per year. ▫ How much is the van worth now? A(1 r )x 18,500(1 0.09)6 10,505.58 ▫ How much was the van worth last year? 18,500(1 0.09)5 11,544.59 ▫ How much was it worth 2 years ago? 18,500(1 0.09)6 2 18,500(1 0.09)4 12,686.37