Lesson 12: Meaning of Exponents

Let’s see how exponents show repeated multiplication.

12.1: Notice and Wonder: Dots and Lines

What do you notice? What do you wonder?

A figure of a series of dot branches. In the center is a black dot. Three branches extend from the black dot with one red dot at the end of each branch. There are three branches that extend from each red dot with one green dot at the end of each branch. There are three branches that extend from each green dot with one yellow dot at the end of each branch. There are three branches that extend from each yellow dot with one blue dot at the end of each branch.

12.2: The Genie’s Offer

You are walking along and you find a brass bottle that looks really, really old. There appears to be some writing on the bottle! You try to clean off some dirt to read it better. A genie appears! He is so happy to be free. He wants to repay you. He offers two ways to repay you and you must choose one:

  • He will give you \$50,000; or
  • He will give you one magical \$1 coin. The magic coin will turn into 2 coins on the first day. The 2 coins will turn into 4 coins on the second day. On the third day, the 4 coins will magically turn into 8 coins. The genie explains that the magic coins will continue this pattern for 28 days.
  1. The number of coins on the third day will be $2 \boldcdot 2 \boldcdot 2$. Can you write another expression using exponents for the number of coins there will be on the third day?
  2. What do $2^5$ and $2^6$ represent in this situation? Evaluate $2^5$ and $2^6$ without a calculator.
  3. How many days would it take for the number of magical coins to exceed \$50,000?
  4. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.

Explore the applet. (Why do you think it stops?)

GeoGebra Applet nfdBbPR3

12.3: Make 81

  1. Here are some expressions. All but one of them equals 16. Find the one that is not equal to 16 and explain how you know.

    $2^3\boldcdot 2$




  2. Write three expressions containing exponents so that each expression equals 81.


When we write an expression like $2^n$, we call $n$ the exponent.

If $n$ is a positive whole number, it tells how many factors of 2 we should multiply to find the value of the expression. For example, $2^1=2$, and $2^5=2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$.

There are different ways to say $2^5$. We can say “two raised to the power of five” or “two to the fifth power” or just “two to the fifth.” 

Practice Problems ▶