Lesson 5 The Power of Negative Thinking Practice Understanding

Learning Focus

Change the form of algebraic expressions using properties of exponents.

How do the properties of exponents help explain methods for changing the form of algebraic expressions, including fractions?

Open Up the Math: Launch, Explore, Discuss

In “Twin Powers,” we learned how negative exponents work with numbers and thought about how they fit into exponential functions. In this lesson, we will be working more with the exponent rules and negative exponents, using variables. The good news is that variables work the same way as numbers. The bad news? There is no bad news.

Sometimes you will be given just the problem, sometimes you’ll be given the steps for rewriting the expression, and sometimes you’ll be given the answer. The purpose of the lesson is for you to get good at working with exponents, so the answers are sometimes provided so that you can be sure that you’re right. If you find that your work is not leading you to the right answer, try rethinking your strategy.

One of the conventions in the math world is to rewrite expressions so they do not have negative exponents. One way to get started is to rewrite the negative exponents as positive in the first step, but you can work with each expression in the way you choose. In each step, write a justification for the step. Problem 1 is entirely worked for you as an example, and the properties of exponents are provided below:

$a^{m} \cdot a^{n} = a^{m + n}$
${(a^{m})}^{n} = a^{m n}$
${(a b)}^{n} = a^{n} b^{n}$
${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$
$\frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0$
$a^{- n} = \frac{1}{a^{n}}$

1.

	Problem:	Step 1:	Step 2:	Answer:
a.	$\frac{x}{x^{- 3}}$	Rewrite with positive exponents. $x \cdot x^{3}$ Because $x^{3} = \frac{1}{x^{- 3}}$	Apply the exponent rules. $x^{1 + 3}$ Exponent multiplication rule	$x^{4}$
b.	$\frac{y^{- 3}}{y^{- 1}}$	Rewrite with positive exponents.	Apply the exponent rules.	$\frac{1}{y^{2}}$
c.	$\frac{3 x^{- 9}}{6 x^{- 11}}$			$\frac{x^{2}}{2}$

Problem:

Step 1:

Step 2:

Answer:

$\frac{x}{x^{- 3}}$

Rewrite with positive exponents.

$x \cdot x^{3}$

Because $x^{3} = \frac{1}{x^{- 3}}$

Apply the exponent rules.

$x^{1 + 3}$

Exponent multiplication rule

$x^{4}$

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

Rewrite with positive exponents.

Apply the exponent rules.

$\frac{1}{y^{2}}$

$\frac{3 x^{- 9}}{6 x^{- 11}}$

$\frac{x^{2}}{2}$

Pause and Reflect

2.

	Problem:	Step 1:	Step 2:	Step 3:
a.	$\frac{a^{2} b^{- 3}}{a^{3} b^{- 1}}$
b.	${(\frac{2 x}{y})}^{- 5}$	Raise everything to the $- 5$ power.	Rewrite with positive exponents.	Evaluate.
c.	${(\frac{c^{4} d}{c^{- 2}})}^{- 4}$

Now we have a challenge: The answer to a problem is given in the last column. Your job is to create “the problem.” The problem must be equivalent to the answer given and contain negative exponents. After writing the problem, show how to rewrite it to get the answer. Challenge yourself to come up with the trickiest problems that you can!

3.

	Problem:	Step 1:	Step 2:	Step 3:	Answer:
a.					$\frac{3 d^{7}}{25 c^{6}}$
b.					$\frac{x^{2} z^{7}}{y^{5}}$
c.					$1$

Ready for More?

Here’s a challenge for you: Combine the following terms and rewrite the expression without negative exponents or common factors.

$\frac{6}{x^{2}} + \frac{5 x^{- 2}}{3^{- 3}}$

Takeaways

Strategies for changing the form of exponential expressions:

Lesson Summary

In this lesson, we learned how to change the form of complicated exponential expressions using the properties of exponents. It is often useful and conventional to write algebraic expressions without negative exponents, which can be done by applying the definition of a negative exponent.

Retrieval

1.

The first and fifth terms of a sequence are given. Use what you know about arithmetic and geometric sequences to fill in the missing numbers. Identify the constant difference for the arithmetic sequences and the common ratio for the geometric sequences. Then write the explicit and recursive equations for each sequence.

Arithmetic	$4$				$324$
Geometric	$4$				$324$

2.

Write the explicit equation that describes the function represented in the table.

$x$	$y$
$- 3$	$- 21$
$- 2$	$- 17$
$- 1$	$- 13$
$0$	$- 9$
$1$	$- 5$
$2$	$- 1$
$3$	$3$

3.

Write the explicit equation that describes the function represented in the graph.