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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.

5.1 Number Talk: What's That Exponent?

Solve each equation mentally.

\frac{100}{1} = 10^x

\frac{100}{x} = 10^1

\frac{x}{100} = 10^0

\frac{100}{1,\!000} = 10^{x}

5.2 Negative Exponent Table

Complete the table to explore what negative exponents mean.

An 8 column table with 3 rows of data. The first column contains a row header for each row. The data are as follows. Row 1: using exponents, 10 cubed; 10 squared; 10 to the first power; blank; blank; blank; blank. Row 2: as a decimal, 1000 point 0; blank; blank; 1 point 0; blank, 0 point 0 1; blank. Row 3: as a fraction, blank, the fraction 100 over 1; blank; the fraction 1 over 1; blank; blank; the fraction 1 over 1000. Above the table are arrows pointing from the 8th column to the 7th, the 7th column to the 6th, and so on. These arrows are labeled "mulitply by 10". Below the table are arrows pointing from the 2nd column to the 3rd, the 3rd column to the 4th, and so on. These arrows are labeled "mulitply by question mark".
  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 10^{\text -7} as a fraction.
  4. Use the patterns you found in the table to write 10^{\text -5} as a decimal.
  5. Write \frac{1}{100,000,000} using a single exponent.
  6. Use the patterns in the table to write 10^{\text -n} as a fraction.

5.3 Follow the Exponent Rules

    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 (10^2)^{\text-3} 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 \frac{10^{\text -2}}{10^{\text -5}} 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 10^{\text -4} \boldcdot 10^3 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 \frac{1}{10} , the exponent decreases by 1: 10^8 \boldcdot \frac{1}{10} = 10^7 This is true for any positive power of 10. We can reason in a similar way that multiplying by 2 factors that are \frac{1}{10} decreases the exponent by 2: \left(\frac{1}{10}\right)^2 \boldcdot 10^8 = 10^6

That means we can extend the rules to use negative exponents if we make 10^{\text-2} = \left(\frac{1}{10}\right)^2 . Just as 10^2 is two factors that are 10, we have that  10^{\text-2} is two factors that are \frac{1}{10} . More generally, the exponent rules we have developed are true for any integers n and m  if we make 10^{\text-n} = \left(\frac{1}{10}\right)^n = \frac{1}{10^n}

Here is an example of extending the rule \frac{10^n}{10^m} = 10^{n-m} to use negative exponents: \frac{10^3}{10^5} = 10^{3-5} = 10^{\text-2}  To see why, notice that \frac{10^3}{10^5} = \frac{10^3}{10^3 \boldcdot 10^2} = \frac{10^3}{10^3} \boldcdot \frac{1}{10^2} =  \frac{1}{10^2} which is equal to 10^{\text-2} .

Here is an example of extending the rule  \left(10^m\right)^n = 10^{m \boldcdot n}  to use negative exponents: \left(10^{\text-2}\right)^{3} = 10^{(\text-2)(3)}=10^{\text-6} To see why, notice that 10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10} . This means that \left(10^{\text-2}\right)^{3} =\left( \frac{1}{10} \boldcdot \frac{1}{10}\right)^3 = \left(\frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot \frac{1}{10}\right)\boldcdot \left(\frac{1}{10}\boldcdot \frac{1}{10}\right) = \frac{1}{10^6} = 10^{\text-6}

Lesson 5 Practice Problems

  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}

  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}

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

    1. (10,\!000)^{\text-1}
    2. (\text{-}10,\!000)
    3. (100)^{\text-2}
    4. (10)^{\text-4}
    5. (\text{-}10)^2

  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:

    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 ABC is similar to EDC . 

      Right triangle ACB and right triangle DEF meet at point C. Line AB is 10. Line AC is 26. Line DC is 36. Line EC is 39. Lines BC and DE are undetermined.
    2. Find the missing side lengths.