7.2.4: Negative Exponents with Powers of 10

Lesson

Let's see what happens when exponents are negative.

Exercise \(\PageIndex{1}\): Number Talk: What's That Exponent?

Solve each equation mentally.

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

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

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

\(\frac{10}{1000}=10^{x}\)

Exercise \(\PageIndex{2}\): Negative Exponent Table

Complete the table to explore what negative exponents mean.

Figure \(\PageIndex{1}\): A table with three rows. Top row, using exponents, 10 cubed, 10 squared, 10 to the first power, blank, blank, blank, blank. Second row, as a decimal, 1000 point 0, blank, blank, 1 point 0, blank, 0 point 0 1, blank. Third row, as a fraction, blank, fraction 100 over 1, blank, fraction 1 over 1, blank, blank, fraction 1 over one thousand. Above the row, arrows point from column to column, right to left, each labeled times 10. Below the bottom row, arrows point from column to column, left to right, each labeled time 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 a multiplier of 10 affect the placement of the decimal in the product? How does the other multiplier affect the placement of the decimal in the product?
3. Use the patterns you found in the table to write \(10^{-7}\) as a fraction.
4. Use the patterns you found in the table to write \(10^{-5}\) as a decimal.
5. Write \(\frac{1}{100,000,000}\) using a single exponent.
6. Use the patterns in the table to write \(10^{-n}\) as a fraction.

Exercise \(\PageIndex{3}\): Follow the Exponential Rules

1. 1. Match each exponential expression with an equivalent multiplication expression:
      \[\begin{array}{lcl}{(10^{2})^{3}}&{\qquad}&{\frac{1}{(10\cdot 10}\cdot \frac{1}{(10\cdot 10}\cdot \frac{1}{(10\cdot 10} }\\{(10^{2})^{-3}}&{\qquad }&{\left(\frac{1}{10}\cdot\frac{1}{10}\right) \left(\frac{1}{10}\cdot\frac{1}{10}\right) \left(\frac{1}{10}\cdot\frac{1}{10}\right) }\\{(10^{-2})^{3}}&{\qquad }&{\frac{1}{\frac{1}{10}\cdot\frac{1}{10}}\cdot \frac{1}{\frac{1}{10}\cdot\frac{1}{10}}\cdot \frac{1}{\frac{1}{10}\cdot\frac{1}{10}} }\\{(10^{-2})^{-3}}&{\qquad}&{(10\cdot 10)(10\cdot 10)(10\cdot 10)}\end{array}\nonumber\]
   2. Write \((10^{2})^{-3}\) as a power of 10 with a single exponent. Be prepared to explain your reasoning.
2. 1. Match each exponential expression with an equivalent multiplication expression:
      \[\begin{array}{lcl}{\frac{10^{2}}{10^{5}}}&{\qquad}&{\frac{\frac{1}{10}\cdot\frac{1}{10}}{\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}}}\\{\frac{10^{2}}{10^{-5}}}&{\qquad}&{\frac{10\cdot 10}{10\cdot 10\cdot 10\cdot 10\cdot 10}}\\{\frac{10^{-2}}{10^{5}}}&{\qquad}&{\frac{\frac{1}{10}\cdot\frac{1}{10}}{10\cdot 10\cdot 10\cdot 10\cdot 10}}\\{\frac{10^{-2}}{10^{-5}}}&{\qquad}&{\frac{10\cdot 10}{\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}}}\end{array}\nonumber\]
   2. Write \(\frac{10^{-2}}{10^{-5}}\) as a power of 10 with a single exponent. Be prepared to explain your reasoning.
3. 1. Match each exponential expression with an equivalent multiplication expression:
   2. \[\begin{array}{lcl}{10^{4}\cdot 10^{3}}&{\qquad}&{(10\cdot 10\cdot 10\cdot 10)\cdot \left(\frac{1}{10}\cdot \frac{1}{10}\cdot \frac{1}{10}\right)}\\{10^{4}\cdot 10^{-3}}&{\qquad }&{\left(\frac{1}{10}\cdot \frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\right)\cdot\left(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\right)}\\{10^{-4}\cdot 10^{3}}&{\qquad }&{\left(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\frac{1}{10}\right)\cdot (10\cdot 10\cdot 10) }\\{10^{-4}\cdot 10^{-3}}&{\qquad}&{(10\cdot 10\cdot 10\cdot 10)\cdot (10\cdot 10\cdot 10)}\end{array}\nonumber\]
   3. Write \(10^{-4}\cdot 10^{3}\) 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 stand in a circle 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?

Summary

When we multiply a positive power of 10 by \(\frac{1}{10}\), the exponent decreases by 1:\(10^{8}\cdot\frac{1}{10}=10^{7}\) 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: \(\left(\frac{1}{10}\right)^{2}\cdot 10^{8}=10^{6}\)

That means we can extend the rules to use negative exponents if we make \(10^{-2}=\left(\frac{1}{10}\right)^{2}\). Just as \(10^{2}\) is two factors that are 10, we have that \(10^{-2}\) 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 \(10^{-n}=\left(\frac{1}{10}\right)^{n}=\frac{1}{10^{n}}\)

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

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

Glossary Entries

Definition: Base (of an exponent)

In expressions like \(5^{3}\) and \(8^{2}\), the 5 and the 8 are called bases. They tell you what factor to multiply repeatedly. For example, \(5^{3}=5\cdot 5\cdot 5\), and \(8^{2}=8\cdot 8\).

Practice

Exercise \(\PageIndex{4}\)

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

1. \(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\)
2. \(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot \frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\)
3. \(\left(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\right)^{2}\)
4. \(\left(\frac{1}{10}\cdot\frac{1}{10}\cdot\frac{1}{10}\right)^{3}\)
5. \((10\cdot 10\cdot 10)^{-2}\)

Exercise \(\PageIndex{5}\)

Write each expression as a single power of 10.

1. \(10^{-3}\cdot 10^{-2}\)
2. \(10^{4}\cdot 10^{-1}\)
3. \(\frac{10^{5}}{10^{7}}\)
4. \(\left(10^{-4}\right) ^{5}\)
5. \(10^{-3}\cdot 10^{2}\)
6. \(\frac{10^{-9}}{10^{5}}\)

Exercise \(\PageIndex{6}\)

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

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

Exercise \(\PageIndex{7}\)

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

Equations:

1. \(y=3x\)
2. \(\frac{1}{2}x=y\)
3. \(y=3.5x\)
4. \(y=\frac{5}{2}x\)

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 3.1.2)

Exercise \(\PageIndex{8}\)

1. Explain why triangle \(ABC\) is similar to \(EDC\).

Figure \(\PageIndex{2}\)

2. Find the missing side lengths.

(From Unit 2.2.3)