Lesson 8

Solve each equation. Be prepared to explain your reasoning.

1. \(\frac {x}{3}=12\)
2. \(3x^2=12\)
3. \(x^3=12\)
4. \(\sqrt[3]{x}=12\)
5. \(\sqrt{3x}=12\)
6. \(\frac {3}{x}=12\)

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.

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?

Complete the tables.

\(x\)			-1	0	\(\frac12\)	1			5
\(2^x\)	\(\frac{1}{32}\)	\(\frac14\)	\(\frac12\)				4	16		256	1,024

\(x\)				\(\frac13\)	\(\frac12\)
\(5^x\)	\(\frac{1}{25}\)	\(\frac15\)	1			5	125	625	3,125

Be prepared to explain how you found the missing values.

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.