# Lesson 5

Negative Exponents with Powers of 10

Let’s see what happens when exponents are negative.

### Problem 1

Write with a single exponent: (ex: $$\frac{1}{10} \boldcdot \frac{1}{10} = 10^{\text-2}$$)

1. $$\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}$$
2. $$\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}$$
3. $$(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10})^2$$
4. $$(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10})^3$$
5. $$(10 \boldcdot 10 \boldcdot 10)^{\text-2}$$

### Problem 2

Write each expression as a single power of 10.

1. $$10^{\text-3} \boldcdot 10^{\text-2}$$
2. $$10^4 \boldcdot 10^{\text-1}$$
3. $$\frac{10^5}{10^7}$$
4. $$(10^{\text-4})^5$$
5. $$10^{\text-3} \boldcdot 10^{\text2}$$
6. $$\frac{10^{\text-9}}{10^5}$$

### Problem 3

Select all of the following that are equivalent to $$\frac{1}{10,000}$$:

A:

$$(10,\!000)^{\text-1}$$

B:

$$(\text{-}10,\!000)$$

C:

$$(100)^{\text-2}$$

D:

$$(10)^{\text-4}$$

E:

$$(\text{-}10)^2$$

### Problem 4

Match each equation to the situation it describes. Explain what the constant of proportionality means in each equation.

Equations:

1. $$y=3x$$
2. $$\frac12x=y$$
3. $$y=3.5x$$
4. $$y=\frac52x$$

Situations:

• A dump truck is hauling loads of dirt to a construction site. After 20 loads, there are 70 square feet of dirt.

• I am making a water and salt mixture that has 2 cups of salt for every 6 cups of water.

• A store has a “4 for \$10” sale on hats.

• For every 48 cookies I bake, my students get 24.

(From Unit 5, Lesson 2.)

### Problem 5

1. Explain why triangle $$ABC$$ is similar to $$EDC$$

2. Find the missing side lengths.
(From Unit 2, Lesson 13.)