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?
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.
 The number of coins on the third day will be \(2 \boldcdot 2 \boldcdot 2\). Write an equivalent expression using exponents.
 What do \(2^5\) and \(2^6\) represent in this situation? Evaluate \(2^5\) and \(2^6\) without a calculator.
 How many days would it take for the number of magical coins to exceed $50,000?
 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

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.
\(2^3\boldcdot 2\)
\(4^2\)
\(\frac{2^5}{2}\)
\(8^2\)

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.”