33.1: Meaning of Exponents
Lesson
Let's see how exponents show repeated multiplication.
Exercise \(\PageIndex{1}\): Notice and Wonder: Dots and Lines
What do you notice? What do you wonder?
Exercise \(\PageIndex{2}\): 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 \(2\cdot 2\cdot 2\). 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?)
Are you ready for more?
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?
Exercise \(\PageIndex{3}\): 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.
\(2^{3}\cdot 2\qquad 4^{2}\qquad\frac{2^{5}}{2}\qquad 8^{2}\)
- 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 \(2^{5}=2\cdot 2\cdot 2\cdot 2\cdot 2\).
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
Exercise \(\PageIndex{4}\)
Select all the expressions that are equivalent to \(64\).
- \(2^{6}\)
- \(2^{8}\)
- \(4^{3}\)
- \(8^{2}\)
- \(16^{4}\)
- \(32^{2}\)
Exercise \(\PageIndex{5}\)
Select all the expressions that equal \(3^{4}\).
- \(7\)
- \(4^{3}\)
- \(12\)
- \(81\)
- \(64\)
- \(9^{2}\)
Exercise \(\PageIndex{6}\)
\(4^{5}\) is equal to 1,024. Evaluate each expression.
- \(4^{6}\)
- \(4^{4}\)
- \(4^{3}\cdot 4^{2}\)
Exercise \(\PageIndex{7}\)
\(6^{3}=216\). Using exponents, write three more expressions whose value is \(216\).
Exercise \(\PageIndex{8}\)
Find two different ways to rewrite \(3xy+6yz\) using the distributive property.
(From Unit 6.2.6)
Exercise \(\PageIndex{9}\)
Solve each equation.
\(a-2.01=5.5\)
\(b+2.01=5.5\)
\(10c=13.71\)
\(100d=13.71\)
(From Unit 6.1.5)
Exercise \(\PageIndex{10}\)
Which expressions represent the total area of the large rectangle? Select all that apply.
- \(6(m+n)\)
- \(6n+m\)
- \(6n+6m\)
- \(6mn\)
- \((n+m)6\)
(From Unit 6.2.5)
Exercise \(\PageIndex{11}\)
Is each statement true or false? Explain your reasoning.
- \(\frac{45}{100}\cdot 72=\frac{45}{72}\cdot 100\)
- \(16\)% of \(250\) is equal to \(250\)% of \(16\)
(From Unit 3.4.7)