Lesson 13
Meaning of Exponents
Let’s see how exponents show repeated multiplication.
13.1: Notice and Wonder: Dots and Lines
What do you notice? What do you wonder?
![A figure of a series of dot branches.](https://staging-cms-im.s3.amazonaws.com/Acgb5THRvTFeTnJiXrHLKtpU?response-content-disposition=inline%3B%20filename%3D%226-6.6.A1.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A1.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T115346Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e371a56671eea473bcff8960fe0cad2eba939cd7d0fd4e9e4f74d233dc68e7f4)
13.2: 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?
13.3: Make 81
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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.
\(2^3\boldcdot 2\)
\(4^2\)
\(\frac{2^5}{2}\)
\(8^2\)
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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.”
Glossary Entries
- cubed
We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).
- exponent
In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).
- squared
We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).