# Lesson 8

Unknown Exponents

• Let’s find unknown exponents.

### 8.1: A Bunch of $x$’s

Solve each equation. Be prepared to explain your reasoning.

1. $$\frac {x}{3}=12$$
2. $$3x^2=12$$
3. $$x^3=12$$
4. $$\sqrt{x}=12$$
5. $$\sqrt{3x}=12$$
6. $$\frac {3}{x}=12$$

### 8.2: A Tessellated Trapezoid

Here is a pattern showing a trapezoid being successively decomposed into four similar trapezoids at each step.

1. If $$n$$ is the step number, how many of the smallest trapezoids are there when $$n$$ is 4? What about when $$n$$ is 10?
2. At a certain step, there are 262,144 smallest trapezoids.
1. Write an equation to represent the relationship between $$n$$ and the number of trapezoids in that step.
2. Explain to a partner how you might find the value of that step number.

### 8.3: Successive Splitting

In a lab, a colony of 100 bacteria is placed on a petri dish. The population triples every hour.

1. How would you estimate or find the population of bacteria in:
1. 4 hours?
2. 90 minutes?
3. $$\frac12$$ hour?
2. How would you estimate or find the number of hours it would take the population to grow to:
1. 1,000 bacteria?
2. double the initial population?

A \$1,000 investment increases in value by 5% each year. About how many years does it take for the value of the investment to double? Explain how you know.

### 8.4: Missing Values

Complete the tables.

 $$x$$ $$2^x$$ -1 0 $$\frac12$$ 1 5 $$\frac{1}{32}$$ $$\frac14$$ $$\frac12$$ 4 16 256 1,024
 $$x$$ $$5^x$$ $$\frac13$$ $$\frac12$$ $$\frac{1}{25}$$ $$\frac15$$ 1 5 125 625 3,125

Be prepared to explain how you found the missing values.

### Summary

Sometimes we know the value of an exponential expression but we don’t know the exponent that produces that value.

For example, suppose the population of a town was 1 thousand. Since then, the population has doubled every decade and is currently at 32 thousand. How many decades has it been since the population was 1 thousand?

If we say that $$d$$ is the number of decades since the population was 1 thousand, then $$1 \boldcdot 2^d$$, or just $$2^d$$, represents the population, in thousands, after $$d$$ decades. To answer the question, we need to find the exponent in $$2^d = 32$$. We can reason that since $$2^5 = 32$$, it has been 5 decades since the population was 1 thousand people.

When did the town have 250 people? Assuming that the doubling started before the population was measured to be 1 thousand, we can write: $$2^d = 0.25$$ or $$2^d = \frac14$$. We know that $$2^{\text-2}=\frac14$$, so the exponent $$d$$ has a value of -2. The population was 250 two decades before it was 1,000.

But it may not always be so straightforward to calculate. For example, it is harder to tell the value of $$d$$ in $$2^d = 805$$ or in $$2^d = 4.5$$. In upcoming lessons, we’ll learn more ways to find unknown exponents.