Lesson 2 Half Interested or More Interesting Solidify Understanding

Jump Start

Notice and Wonder

Previously, you examined the following context:

Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of $60 milligrams$ of anti-parasite medicine is given to a dog and the medicine breaks down such that $20 %$ of the medicine becomes ineffective every hour. How much of the $60 milligram$ dose is still active in the dog’s bloodstream after $3 hours$ , after $4.25 hours$ , and after $n$ hours?

Here are three representations of that context. List at least two things that you notice and one thing you are wondering about relative to these representations.

Representation #1:

Representation #2:

Representation #3:

$A (t) = 60 {(0.8)}^{t}$

Learning Focus

Examine how the properties of exponents work with rational exponents.

Write equivalent exponential functions using different growth factors.

What do rational exponents and negative exponents mean in contexts?

Do the laws of exponents work with rational exponents?

How does the growth factor change if we focus on a month of exponential growth instead of a year?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita, the Martinez twins, have run a summer business every year for the past $5$ years. Their first business, a neighborhood lemonade stand, earned a small profit that their father insisted they deposit in a savings account at the local bank. When the Martinez family moved a few months later, the twins decided to leave the money in the bank where it has been earning $5 %$ interest annually. Carlos was reminded of the money when he found the annual bank statement they had received in the mail.

“Remember how Dad said we could withdraw this money from the bank when we are $20$ years old,” Carlos said to Clarita. “We have $$ 382.88$ in the account now. I wonder how much that will be $5$ years from now?”

1.

Carlos calculates the value of the account $1$ year at a time. He has just finished calculating the value of the account for the first $4$ years. Describe how he can find the next year’s balance and record that value in the table.

Year	Amount
$0$	$382.88$
$1$	$402.02$
$2$	$422.12$
$3$	$443.23$
$4$	$465.39$
$5$

2.

Clarita thinks Carlos is silly calculating the value of the account one year at a time, and says that he could have written a formula for the $n th$ year and then evaluated his formula when $n = 5$ . Write Clarita’s formula for the $n th$ year and use it to find the account balance at the end of year $5$ .

3.

Carlos was surprised that Clarita’s formula gave the same account balance as his year-by-year strategy. Explain, in a way that would convince Carlos, why this is so.

“I can’t remember how much money we earned that summer,” said Carlos. “I wonder if we can figure out how much we deposited in the account five years ago, knowing the account balance now?”

4.

Carlos continued to use his strategy to extend his table year-by-year back $5$ years. Explain what you think Carlos is doing to find his table values one year at a time and continue filling in the table until you get to $- 5$ , which Carlos uses to represent “ $5$ years ago.”

Year	Amount
$- 5$
$- 4$
$- 3$
$- 2$
$- 1$	$364.65$
$0$	$382.88$
$1$	$402.02$
$2$	$422.12$
$3$	$443.23$
$4$	$465.39$
$5$

Explanation:

5.

Clarita evaluated her formula for $n = - 5$ . Again, Carlos is surprised that they get the same results. Explain why Clarita’s method works.

Clarita doesn’t think leaving the money in the bank for another $5$ years is such a great idea and suggests that they invest the money in their next summer business. “We’ll have some start-up costs, and this will pay for them without having to withdraw money from our other accounts.”

Carlos remarked, “But we’ll be withdrawing our money halfway through the year. Do you think we’ll lose out on this year’s interest?”

“No, they’ll pay us a half-year portion of our interest,” replied Clarita.

“But how much will that be?” asked Carlos.

6.

Calculate the account balance and how much interest you think Carlos and Clarita should be paid if they withdraw their money $\frac{1}{2}$ year from now. Remember that they currently have $$ 382.88$ in the account and that they earn $5 %$ annually. Describe your strategy.

Clarita used this strategy: She substituted $\frac{1}{2}$ for $n$ in the formula $A = 382.88 {(1.05)}^{n}$ and recorded this as the account balance.

Carlos had some questions about Clarita’s strategy:

What numerical amount do we multiply by when we use ${(1.05)}^{\frac{1}{2}}$ as a factor?
What happens if we multiply by ${(1.05)}^{\frac{1}{2}}$ and then multiply the result by ${(1.05)}^{\frac{1}{2}}$ again? Shouldn’t that be a full year’s worth of interest? Is it?
If multiplying by ${(1.05)}^{\frac{1}{2}} \cdot {(1.05)}^{\frac{1}{2}}$ is the same as multiplying by $1.05$ , what does that suggest about the value of ${(1.05)}^{\frac{1}{2}}$ ?

7.

Answer each of Carlos’s questions listed as best as you can.

Pause and Reflect

As Carlos is reflecting on this work, Clarita notices the date on the bank statement that started this whole conversation. “This bank statement is three months old!” she exclaims. “That means the bank will owe us $\frac{3}{4}$ of a year’s interest.”

“So how much interest will the bank owe us then?” asked Carlos.

8.

Find as many ways as you can to answer Carlos’s question: How much will their account be worth in $\frac{3}{4}$ of a year (nine months) if it earns $5 %$ annually and is currently worth $$ 382.88$ ?

Carlos now knows he can calculate the amount of interest earned on an account in smaller increments than one full year. He would like to determine how much money is in an account each month that earns $5 %$ annually with an initial deposit of $$ 300$ .

He starts by considering the amount in the account each month during the first year. He knows that by the end of the year the account balance should be $$ 315$ , since it increases $5 %$ during the year.

9.

Complete the table showing what amount is in the account each month during the first $12$ months.

Deposit	$\frac{1}{12}$	$\frac{2}{12}$	$\frac{3}{12}$	$\frac{4}{12}$	$\frac{5}{12}$	$\frac{6}{12}$	$\frac{7}{12}$	$\frac{8}{12}$	$\frac{9}{12}$	$\frac{10}{12}$	$\frac{11}{12}$	$1 year$
$$ 300$												$$ 315$

10.

What number did you multiply the account by each month to get the next month’s balance?

Carlos knows the exponential equation that gives the account balance for this account on an annual basis is $A = 300 {(1.05)}^{t}$ . Based on his work finding the account balance each month, Carlos writes the following equation for the same account: $A = 300 {({1.05}^{\frac{1}{12}})}^{12 t}$ .

11.

Verify that both equations give the same results. Using the properties of exponents, explain why these two equations are equivalent.

12.

What is the meaning of the $12 t$ in this equation?

Carlos shows his equation to Clarita. She suggests his equation could also be approximated by $A = 300 {(1.004)}^{12 t}$ , since ${1.05}^{\frac{1}{12}} = 1.004$ . Carlos replies, “I know the $1.05$ in the equation $A = 300 {(1.05)}^{t}$ means I am earning $5 %$ interest annually, but what does the $1.004$ mean in your equation?”

13.

Answer Carlos’s question. What does the $1.004$ mean in $A = 300 {(1.004)}^{12 t}$ ?

Pause and Reflect

The properties of exponents can be used to explain why ${[{(1.05)}^{\frac{1}{12}}]}^{12 t} = {1.05}^{t}$ . Here are some more examples of using the properties of exponents with rational exponents. For each of the following, rewrite the expression using the properties of exponents and explain what the expression means in terms of the context.

14.

${(1.05)}^{\frac{1}{12}} \cdot {(1.05)}^{\frac{1}{12}} \cdot {(1.05)}^{\frac{1}{12}}$

15.

${[{(1.05)}^{\frac{1}{12}}]}^{6}$

16.

${(1.05)}^{\frac{- 1}{12}}$

17.

${(1.05)}^{2} \cdot {(1.05)}^{\frac{1}{4}}$

18.

$\frac{{(1.05)}^{2}}{{(1.05)}^{\frac{1}{2}}}$

Ready for More?

Use ${[{(1.05)}^{\frac{1}{12}}]}^{12} = 1.05$ to explain why ${(1.05)}^{\frac{1}{12}} = \sqrt[12]{1.05}$ .

Takeaways

The following properties of exponents that make sense for positive integer exponents also apply and make sense for negative integer exponents and for fractional exponents:

We can interpret the negative exponential factor in $a b^{- x}$ as meaning:

We can interpret a fractional exponential factor in $a b^{\frac{m}{n}}$ as meaning:

Lesson Summary

In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents.

Retrieval

Use the rules of exponents to find three expressions that would be equivalent to the one provided.

1.

$5^{7}$

2.

$3^{- 4}$

Rewrite each of the expressions.

3.

$x^{a} \cdot x^{b}$

4.

$\frac{y^{5} \cdot y^{3}}{y^{6}}$

5.

$\frac{{(n^{2})}^{5}}{n^{7}}$