Lesson 2 Twin Powers Solidify Understanding

Jump Start

Illustrate with an example why each of the following properties of exponents works:

1.

$a^{m} \cdot a^{n} = a^{m + n}$

2.

${(a^{m})}^{n} = a^{m n}$

3.

$\frac{a^{m}}{a^{n}} = a^{m - n}$

4.

${(a \cdot b)}^{m} = a^{m} \cdot b^{m}$

5.

${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$

Apply rule 3 to complete these statements:

6.

By rule 3, $\frac{a^{3}}{a^{3}} = \underset{―}{}$ . By dividing out common factors in fractions, $\frac{a^{3}}{a^{3}} = \underset{―}{}$ .

Therefore, $\underset{―}{}$ .

7.

By rule 3, $\frac{a^{3}}{a^{4}} = \underset{―}{}$ . By dividing out common factors in fractions, $\frac{a^{3}}{a^{4}} = \underset{―}{}$ .

Therefore, $\underset{―}{}$ .

Learning Focus

Relate the key features of exponential functions to properties of negative exponents.

Rewrite exponential expressions that involve negative exponents.

How does my understanding of the properties of exponents help me explain the key features of exponential functions?

How does my understanding of the properties of exponents help me rewrite exponential expressions that contain negative exponents?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita have been working on their math together and noticed something interesting.

Carlos: I’ve been looking at the graph of $f (x) = 2^{x}$ on my calculator and trying to make sense of the graph when the inputs are negative. Here’s what I see on the graph:

1.

What observations do you make about Carlos’s graph, especially when the $x$ -values are negative?

Clarita: Sometimes it helps to make a table and look for patterns in the numbers. I’m going to try that.

2.

Complete Clarita’s table, making sure that your table matches the graph. Record any patterns that you see. Use fractions instead of decimals throughout the task to make the patterns easier to see.

$x$	$4$	$3$	$2$	$1$	$0$	$- 1$	$- 2$	$- 3$	$- 4$
$f (x) = 2^{x}$	$16$	$8$	$4$	$2$	$1$

Carlos: So, here’s what I think is really strange. Look at the graph of $g (x) = {(\frac{1}{2})}^{x}$ .

3.

Compare the graphs of $g (x) = {(\frac{1}{2})}^{x}$ and $f (x) = 2^{x}$ . What similarities and differences do you notice?

Clarita: I think we should make a table to compare the two functions. I’ve added a column to my previous table.

4.

a.

Complete Clarita’s table and make sure that the table matches the graphs of the functions.

$x$	$4$	$3$	$2$	$1$	$0$	$- 1$	$- 2$	$- 3$	$- 4$
$f (x) = 2^{x}$	$16$	$8$	$4$	$2$	$1$
$g (x) = {(\frac{1}{2})}^{x}$

b.

What similarities and differences do you notice in the table values?

How would you explain the similarities and differences that you have observed?

Carlos: I remember this about negative exponents: $a^{- n} = \frac{1}{a^{n}}$ . Negative exponents always have a “twin”: an equivalent expression that is written with a positive exponent.

5.

Explain how what Carlos remembers about negative exponents relates to the relationships you have observed in $g (x) = {(\frac{1}{2})}^{x}$ and $f (x) = 2^{x}$ or to the work with negative exponents in the previous task.

Clarita: I think we could play around with these negative exponents and use them with bases other than $2$ .

6.

Try applying the multiplication rule for exponents, $a^{m} \cdot a^{n} = a^{m + n}$ , to these expressions. When you have finished, use another method to verify that your answers are correct.

a.

$2^{5} \cdot 2^{- 3}$

b.

$3^{2} \cdot 3^{- 2}$

7.

Try applying the division rule for exponents, $\frac{a^{m}}{a^{n}} = a^{m - n}$ , to these expressions. When you have finished, use another method to verify that your answers are correct.

a.

$\frac{3^{2}}{3^{- 1}}$

b.

$\frac{5^{- 2}}{5^{3}}$

8.

Try applying the power of a power rule, ${(a^{m})}^{n} = a^{m n}$ , for exponents to these expressions. When you have finished, use another method to verify that your answers are correct.

a.

${(3^{3})}^{- 2}$

b.

${(5^{- 1})}^{- 2}$

Carlos: All of this work with negative exponents has made me realize something—I can write two different exponential rules for the same function!

9.

Carlos has already written the rule $g (x) = {\frac{1}{2}}^{x}$ for this function. What is the other rule he is thinking about?

10.

Can you also write two different rules for this function?

Clarita: Wow, that’s interesting. So how can I tell if an exponential function is always increasing or always decreasing before I graph it?

11.

How would you answer Clarita’s question?

Ready for More?

Let’s think about negative exponents with negative bases. Find values for the following:

a.

${(- 3)}^{- 2}$

b.

${(- 3)}^{- 3}$

c.

$- 3^{- 2}$

d.

What would happen if we try to use $- 3$ as the base of an exponential function? Plot a few points on the graph of $f (x) = {(- 3)}^{x}$ . Be sure to try positive and negative integers, as well as fractions, for $x$ .

Takeaways

I noticed some useful strategies for working with expressions that include negative exponents, such as:

I noticed several things about exponential functions over the past few days, and each of these observations can be supported by the properties of exponents:

The end behavior on one end of an exponential function approaches the $x$ -axis as an asymptote:

Vocabulary

asymptote
horizontal asymptote
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we noticed several characteristics of the graphs and tables of exponential functions that can be explained using our understanding of negative exponents. We also used the rules of exponents to change the form of numeric expressions that contain negative exponents.

Retrieval

Rewrite the radicals.

1.

$\sqrt{18}$

2.

$\sqrt[3]{24}$

3.

$\sqrt[5]{64}$

4.

$\sqrt{75}$

Use the function rules to find the indicated values.

5.

$f (x) = 5 x - 8$

a.

Find $f (3)$ .

b.

Find $f (x) = 27$ .

c.

Find $f (- 2)$ .

6.

$g (x) = 3 {(4)}^{x}$

a.

Find $g (4)$ .

b.

Find $g (x) = 12$ .

c.

Find $g (- 1)$ .