6.5 Investigation The Division Property of Exponents Name ______
Step 1: Write the numerator and the denominator of each quotient in expanded form. Then
simplify to eliminate common factors. Rewrite the factors that remain with exponents.
Use your calculator to check your answers.
a) 5956 b) 33 ∙ 533 ∙ 52 c) 44x642x3
Step 2: Compare the exponents in each final expression you got in Step 1 to the exponents in the
quotient. Describe a way to find the exponents in the final expression without using
expanded form.
Step 3: Use your method from Step 2 to rewrite this expression so that it is not a fraction. You
can leave 0.0812 as a fraction.
515(1+0.0812)24511(1+0.0812)18
Recall that exponential growth is related to repeated multiplication. When you look ahead in time you multiply by repeated constant multipliers, 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.
a) After 7 years the balance in a savings account is 500(1 + 0.04)7. What does the expression 500(1+0.04)7(1+0.04)3 mean in this situation? Rewrite this expression with a single exponent.
b) After 9 years of depreciation, the value of a car is 21,300(1 – 0.12)9. What does the expression 21,300(1- 0.12)9(1- 0.12)5 mean in this situation? Rewrite this expression with a single exponent.
c) 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.
d) The expression A(1 + r)n can model n time periods of exponential growth. What expression models the growth m time periods earlier?
Step 5: How does looking back in time with an exponential model relate to dividing expressions
with exponents?
Step 6 Simplify each expression.
*hint*( When dividing you divide the numbers and subtract the exponents)
a) b) c)
Step 7 Six years ago, Anne bought a van for $18,500. She estimates it will depreciate about
9% per year. Write an expression and then solve the following:
Expression ______
a) How much is the van worth now?
b) How much was it worth last year?
c) How much was it worth two years ago?
d) In what year will the van be less than $500?
Step 8 Write a recursive equation that calculates the value of Anne’s van. Let U0 equal the
purchase price of the van.
U0 =
Un = Un-1 *
Step 9 Write an equation using A = P(1 + r)t for the value of the van. Let t = 0 represent the year
she bought the van. (*Hint because the van is depreciating r will be a negative value.)
Step 10 Anne’s Dad has a classic car in the garage that was worth$12,000 the year Anne bought
her van. If the classic car appreciates 5% a year, write an equation using A = P(1 + r)t for
the value of the van. Let t = 0 represent the year Anne bought the van.
Step 11 Write a recursive equation that calculates the value of the classic car. Let U0 equal the
value of the car when Anne bought the van.
U0 =
Un = Un-1 *
Step 12 Graph both equations on your calculator and then sketch the graphs. Use the intersect
feature on your calculator to find when the cars would have the same value.
In ______years both cars will be worth $______.