# Chapter 8: Exponents and Exponential Functions

**Basic**Created by: CK-12

## Introduction

Exponential functions occur in daily situations; money in a bank account, population growth, the decay of carbon-14 in living organisms, and even a bouncing ball. Exponential equations involve **exponents**, or the concept of repeated multiplication. This chapter focuses on combining expressions using the properties of exponents. The latter part of this chapter focuses on creating exponential equations and using the models to predict.

- 8.1.
## Exponential Properties Involving Products

- 8.2.
## Exponential Properties Involving Quotients

- 8.3.
## Negative Exponents

- 8.4.
## Fractional Exponents

- 8.5.
## Scientific Notation

- 8.6.
## Scientific Notation with a Calculator

- 8.7.
## Exponential Growth

- 8.8.
## Exponential Decay

- 8.9.
## Geometric Sequences and Exponential Functions

- 8.10.
## Applications of Exponential Functions

### Chapter Summary

## Summary

This chapter begins by talking about exponential properties involving products and quotients. It also discusses zero and negative exponents, as well as fractional exponents. Next, scientific notation is covered, and evaluating scientific notation with a calculator is outlined in detail. Instruction is then given on exponential growth and decay, and geometric sequences and exponential functions are highlighted. Finally, the chapter concludes by giving examples of how to solve real-world problems.

### Exponents and Exponential Functions Review

Define the following words.

- Exponent
- Geometric Sequence

Use the product properties to simplify the following expressions.

- \begin{align*}5 \cdot 5 \cdot 5 \cdot 5\end{align*}
- \begin{align*}(3x^2y^3) \cdot (4xy^2)\end{align*}
- \begin{align*}a^3 \cdot a^5 \cdot a^6\end{align*}
- \begin{align*}(\gamma^3)^5\end{align*}
- \begin{align*}(x \cdot x^3 \cdot x^5)^{10}\end{align*}
- \begin{align*}(2a^3 b^3)^2\end{align*}

Use the quotient properties to simplify the following expressions.

- \begin{align*}\frac{c^5}{c^3}\end{align*}
- \begin{align*}\frac{a^6}{a}\end{align*}
- \begin{align*}\frac{a^5b^4}{a^3b^2}\end{align*}
- \begin{align*}\frac{x^4 y^5 z^2}{x^3 y^2 z}\end{align*}

Simplify the following expressions.

- \begin{align*}\frac{6^5}{6^5}\end{align*}
- \begin{align*}\frac{\gamma^2}{\gamma^5}\end{align*}
- \begin{align*}\frac{7^3}{7^6}\end{align*}
- \begin{align*}\frac{2}{\chi^3}\end{align*}
- \begin{align*}\sqrt[4]{\alpha^3}\end{align*}
- \begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}
- \begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}

Write the following in scientific notation.

- 557,000
- 600,000
- 20
- 0.04
- 0.0417
- 0.0000301
- The distance from the Earth to the moon: 384,403 km
- The distance from Earth to Jupiter: 483,780,000 miles
- According to the CDC, the appropriate level of lead in drinking water should not exceed 15 parts per billion (EPA’s Lead & Copper Rule).

Write the following in standard notation.

- \begin{align*}3.53 \times 10^3\end{align*}
- \begin{align*}89 \times 10^5\end{align*}
- \begin{align*}2.12 \times 10^6\end{align*}
- \begin{align*}5.4 \times 10^1\end{align*}
- \begin{align*}7.9 \times 10^{-3}\end{align*}
- \begin{align*}4.69 \times 10^{-2}\end{align*}
- \begin{align*}1.8 \times 10^{-5}\end{align*}
- \begin{align*}8.41 \times 10^{-3}\end{align*}

Make a graph of the following exponential growth/decay functions.

- \begin{align*}\gamma=3 \cdot (6)^x\end{align*}
- \begin{align*}\gamma=2 \cdot \left(\frac{1}{3}\right)^x\end{align*}
- Marissa was given 120 pieces of candy for Christmas. She ate one-fourth of them each day. Make a graph to find out in how many days Marissa will run out of candy.
- Jacoby is given $1500 for his graduation. He wants to invest it. The bank gives a 12% investment rate each year. Make a graph to find out how much money Jacoby will have in the bank after six years.

Determine what the common ratio is for the following geometric sequences and finish each sequence.

- 1, 3, __, __, 81
- __, 5, __, 125, 625
- 7, __, 343, 2401, __
- 5, 1.5, __, 0.135, __
- The population of ants in Ben’s room increases three times daily. He starts with only two ants. Make a geometric graph to determine how many ants will be in Ben’s room at the end of a 30-day month if he does not take care of the problem.

### Exponents and Exponential Functions Test

Simplify the following expressions.

- \begin{align*}x^3 \cdot x^4 \cdot x^5\end{align*}
- \begin{align*}(a^3)^7\end{align*}
- \begin{align*}(y^2 z^4)^7\end{align*}
- \begin{align*}\frac{a^3}{a^5}\end{align*}
- \begin{align*}\frac{x^3 y^2}{x^6 y^4}\end{align*}
- \begin{align*}\left(\frac{3x^8 y^2}{9x^6 y^5}\right)^3\end{align*}
- \begin{align*}\frac{3^4}{3^4}\end{align*}
- \begin{align*}\frac{2}{x^3}\end{align*}
- \begin{align*}\sqrt[3]{5^6}\end{align*}

Complete the following story problems.

- The intensity of a guitar amp is 0.00002. Write this in scientific notation.
- Cole loves turkey hunting. He already has two after the first day of hunting season. If this number doubles each day, how many turkeys will Cole have after 11 days? Make a table for the geometric sequence.
- The population of a town increases by 20% each year. It first started with 89 people. What will the population of the town be after 15 years?
- A radioactive substance decays 2.5% every hour. What percent of the substance will be left after nine hours?
- After an exterminator comes to a house to exterminate cockroaches, the bugs leave the house at a rate of 16% an hour. How long will it take 55 cockroaches to leave the house after the exterminator comes?

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