# Lesson 15: Equivalent Exponential Expressions

Let's investigate expressions with variables and exponents.

## 15.1: Up or Down?

1. Find the values of $3^x$ and $\left(\frac13\right)^x$ for different values of $x$.
row 1  $x$   $3^x$ $\left(\frac13\right)^x$
row 2 1
row 3 2
row 4 3
row 5 4
1. What patterns do you notice?

## 15.2: What's the Value?

Evaluate each expression for the given value of $x$.

1. $3x^2$ when $x$ is 10

2. $3x^2$ when $x$ is $\frac19$

3. $\frac{x^3}{4}$ when $x$ is 4

4. $\frac{x^3}{4}$ when $x$ is $\frac12$

5. $9+x^7$ when $x$ is 1

6. $9+x^7$ when $x$ is $\frac12$

## 15.3: Exponent Experimentation

Find a solution to each equation in the list that follows. (Numbers in the list may be a solution to more than one equation, and not all numbers in the list will be used.)

1. $64=x^2$
2. $64=x^3$
3. $2^x=32$
4. $x=\left( \frac25 \right)^3$
1. $\frac{16}{9}=x^2$
2. $2\boldcdot 2^5=2^x$
3. $2x=2^4$
4. $4^3=8^x$

List:

$\frac{8}{125}$

$\frac{6}{15}$

$\frac{5}{8}$

$\frac89$

1

$\frac43$

2

3

4

5

6

8

## Summary

In this lesson, we saw expressions that used the letter $x$ as a variable. We evaluated these expressions for different values of $x$.

• To evaluate the expression $2x^3$ when $x$ is 5, we replace the letter $x$ with 5 to get $2 \boldcdot 5^3$. This is equal to $2 \boldcdot 125$ or just 250. So the value of $2x^3$ is 250 when $x$ is 5.
• To evaluate $\frac{x^2}{8}$ when $x$ is 4, we replace the letter $x$ with 4 to get $\frac{4^2}{8} = \frac{16}{8}$, which equals 2. So $\frac{x^2}{8}$ has a value of 2 when $x$ is 4.

We also saw equations with the variable $x$ and had to decide what value of $x$ would make the equation true.

• Suppose we have an equation $10 \boldcdot 3^x = 90$ and a list of possible solutions: ${1, 2, 3, 9, 11}$. The only value of $x$ that makes the equation true is 2 because $10 \boldcdot 3^2 = 10 \boldcdot 3 \boldcdot 3$, which equals 90. So 2 is the solution to the equation.