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.

 

12.2: The Genie’s Offer

You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one:

  • $50,000; or
  • A magical $1 coin. The coin will turn into two coins on the first day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days.
  1. The number of coins on the third day will be \(2 \boldcdot 2 \boldcdot 2\). Write an equivalent expression using exponents.
  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?)



A scientist is growing a colony of bacteria in a petri dish. She knows that the bacteria are growing and that the number of bacteria doubles every hour.

When she leaves the lab at 5 p.m., there are 100 bacteria in the dish. When she comes back the next morning at 9 a.m., the dish is completely full of bacteria. At what time was the dish half full?

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.

    \(2^3\boldcdot 2\)

    \(4^2\)

    \(\frac{2^5}{2}\)

    \(8^2\)

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

Summary

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.”