Lesson 14

Expressions with Exponents

Let's use the meaning of exponents to decide if equations are true.

14.1: Which One Doesn’t Belong: Twos

Which one doesn’t belong?

$$2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$

$$16$$

$$2^4$$

$$4 \boldcdot 2$$

14.2: Is the Equation True?

Decide whether each equation is true or false, and explain how you know.

1. $$2^4=2 \boldcdot 4$$

2. $$3+3+3+3+3=3^5$$

3. $$5^3=5 \boldcdot 5 \boldcdot 5$$

4. $$2^3=3^2$$

5. $$16^1=8^2$$

6. $$\frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 = 4 \boldcdot \frac12$$

7. $$\left( \frac12 \right)^4=\frac{1}{8}$$

8. $$8^2 = 4^3$$

In each list, find expressions that are equivalent to each other and explain to your partner why they are equivalent. Your partner listens to your explanation. If you disagree, explain your reasoning until you agree. Switch roles for each list. (There may be more than two equivalent expressions in each list.)

1. $$5 \boldcdot 5$$
2. $$2^5$$
3. $$5^2$$
4. $$2 \boldcdot 5$$
1. $$4^3$$
2. $$3^4$$
3. $$4 \boldcdot 4 \boldcdot 4$$
4. $$4+4+4$$
1. $$6+6+6$$
2. $$6^3$$
3. $$3^6$$
4. $$3 \boldcdot 6$$
1. $$11^5$$
2. $$11 \boldcdot 11 \boldcdot 11 \boldcdot 11 \boldcdot 11$$
3. $$11 \boldcdot 5$$
4. $$5^{11}$$
1. $$\frac15 \boldcdot \frac15 \boldcdot \frac15$$
2. $$\left( \frac15 \right)^3$$
3. $$\frac{1}{15}$$
4. $$\frac{1}{125}$$
1. $$\left( \frac53 \right)^2$$
2. $$\left( \frac35 \right)^2$$
3. $$\frac{10}{6}$$
4. $$\frac{25}{9}$$

What is the last digit of $$3^{1,000}$$? Show or explain your reasoning.

Summary

When working with exponents, the bases don’t have to always be whole numbers. They can also be other kinds of numbers, like fractions, decimals, and even variables. For example, we can use exponents in each of the following ways:

$$\displaystyle \left(\frac{2}{3}\right)^4 = \frac{2}{3} \boldcdot \frac{2}{3} \boldcdot \frac{2}{3} \boldcdot \frac{2}{3}$$

$$\displaystyle (1.7)^3 = (1.7) \boldcdot (1.7) \boldcdot (1.7)$$

$$\displaystyle x^5 = x \boldcdot x \boldcdot x \boldcdot x \boldcdot x$$

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 $$s^2$$.