# Lesson 8

Unknown Exponents

- Let’s find unknown exponents.

### Problem 1

A pattern of dots grows exponentially. The table shows the number of dots at each step of the pattern.

step number | 0 | 1 | 2 | 3 |
---|---|---|---|---|

number of dots | 1 | 5 | 25 | 125 |

- Write an equation to represent the relationship between the step number, \(n\), and the number of dots, \(y\).
- At one step, there are 9,765,625 dots in the pattern. At what step number will that happen? Explain how you know.

### Problem 2

A bacteria population is modeled by the equation \(p(h) = 10,\!000 \boldcdot 2^h\), where \(h\) is the number of hours since the population was measured.

About how long will it take for the population to reach 100,000? Explain your reasoning.

### Problem 3

Complete the table.

\(x\) | -2 | 0 | \(\frac{1}{3}\) | 1 | ||||
---|---|---|---|---|---|---|---|---|

\(10^x\) | \(\frac{1}{10,000}\) | \(\frac{1}{1,000}\) | \(\frac{1}{100}\) | \(\hspace{.6cm}\) | \(\hspace{.6cm}\) | \(\hspace{.6cm}\) | 1,000 | 1,000,000,000 |

### Problem 4

Here is a graph of \(y = 3^x\).

What is the approximate value of \(x\) satisfying \(3^x = 10,\!000\)? Explain how you know.

### Problem 5

One account doubles every 2 years. A second account triples every 3 years. Assuming the accounts start with the same amount of money, which account is growing more rapidly?

### Problem 6

How would you describe the output of this graph for:

- inputs from 0 to 1
- inputs from 3 to 4

### Problem 7

The half-life of carbon-14 is about 5730 years.

- Complete the table, which shows the amount of carbon-14 remaining in a plant fossil at the different times since the plant died.
- About how many years will it be until there is 0.1 picogram of carbon-14 remaining in the fossil? Explain how you know.

years | picograms |
---|---|

0 | 3 |

5730 | |

\(2 \boldcdot 5730\) | |

\(3 \boldcdot 5730\) | |

\(4 \boldcdot 5730\) |