# Lesson 5Negative Exponents with Powers of 10

Let’s see what happens when exponents are negative.

### Learning Targets:

• I can use the exponent rules with negative exponents.
• I know what it means if 10 is raised to a negative power.

## 5.1Number Talk: What's That Exponent?

Solve each equation mentally.

## 5.2Negative Exponent Table

Complete the table to explore what negative exponents mean.

1. As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?
2. How does each of these multipliers affect the placement of the decimal?
3. Use the patterns you found in the table to write as a fraction.
4. Use the patterns you found in the table to write as a decimal.
5. Write using a single exponent.
6. Use the patterns in the table to write as a fraction.

1. Match the expressions that describe repeated multiplication in the same way:

 \left(10^2\right)^3 \frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)} \left(10^2\right)^{\text -3} \left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) \left(10^{\text -2}\right)^3 \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} } \boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} } \left(10^{\text -2}\right)^{\text-3} (10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)
2. Write as a power of 10 with a single exponent. Be prepared to explain your reasoning.
1. Match the expressions that describe repeated multiplication in the same way:

 \frac{10^2}{10^5} \frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10} } \frac{10^2}{10^{\text -5}} \frac{10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} \frac{10^{\text -2}}{10^5} \frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ 10 \boldcdot 10\boldcdot 10\boldcdot 10\boldcdot 10 } \frac{10^{\text -2}}{10^{\text -5}} \frac{ 10 \boldcdot 10 }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10}}
2. Write as a power of 10 with a single exponent. Be prepared to explain your reasoning.
1. Match the expressions that describe repeated multiplication in the same way:

 10^4 \boldcdot 10^3 (10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot ( \frac{1}{10} \boldcdot  \frac{1}{10}\boldcdot  \frac{1}{10}) 10^4 \boldcdot 10^{\text -3} \left(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot  \frac{1}{10} \boldcdot  \frac{1}{10}\right) 10^{\text -4} \boldcdot 10^3 \left(\frac{1}{10}\boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left(10 \boldcdot 10 \boldcdot 10\right) 10^{\text -4} \boldcdot 10^{\text -3} (10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot (10 \boldcdot 10 \boldcdot 10)
2. Write as a power of 10 with a single exponent. Be prepared to explain your reasoning.

### Are you ready for more?

Priya, Jada, Han, and Diego are playing a game. They stand in a circle in this order and take turns playing a game.

Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.

Priya says, “I knew I’d be the only one left, since I went first.”

1. Record this game on paper a few times with different numbers of players. Does the person who starts always win?
2. Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?

## Lesson 5 Summary

When we multiply a positive power of 10 by , the exponent decreases by 1: This is true for any positive power of 10. We can reason in a similar way that multiplying by 2 factors that are decreases the exponent by 2:

That means we can extend the rules to use negative exponents if we make . Just as is two factors that are 10, we have that  is two factors that are . More generally, the exponent rules we have developed are true for any integers and  if we make

Here is an example of extending the rule to use negative exponents:  To see why, notice that which is equal to .

Here is an example of extending the rule  to use negative exponents: To see why, notice that . This means that

## Lesson 5 Practice Problems

1. Write with a single exponent: (ex: )

2. Write each expression as a single power of 10.

3. Select all of the following that are equivalent to :

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

Equations:

Situations:

1. A dump truck is hauling loads of dirt to a construction site. After 20 loads, there are 70 square feet of dirt.
2. I am making a water and salt mixture that has 2 cups of salt for every 6 cups of water.
3. A store has a “4 for \$10” sale on hats.
4. For every 48 cookies I bake, my students get 24.
1. Explain why triangle is similar to

2. Find the missing side lengths.