# Chapter 6: Exponents and Exponential Functions

Difficulty Level: Advanced Created by: CK-12

## Introduction

Here you'll learn all about exponents in algebra. You will learn the properties of exponents and how to simplify exponential expressions. You will learn how exponents can help you write very large or very small numbers with scientific notation. You will also learn how to solve different types of exponential equations where the variable appears as the exponent or the base. Finally, you will explore different types of exponential functions of the form \begin{align*}y=b^x, y=ab^x, y=ab^{\frac{x}{c}},\end{align*} and \begin{align*}y=ab^{\frac{x}{c}}+d\end{align*} as well as applications of exponential functions.

Chapter Outline

## Summary

You learned that in an expression like \begin{align*}2^x\end{align*}, the "2" is the base and the "x" is the exponent. You learned the following laws of exponents that helped you to simplify expressions with exponents:

• \begin{align*}a^m \times a^n=a^{m+n}\end{align*}
• \begin{align*}\frac{a^m}{a^n}=a^{m-n} \ (\text{if} \ m > n, a \neq 0)\end{align*}
• \begin{align*}(a^m)^n=a^{mn}\end{align*}
• \begin{align*}(ab)^n=a^nb^n\end{align*}
• \begin{align*}\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \ (b \neq 0)\end{align*}
• \begin{align*}a^0=1 \ (a \neq 0)\end{align*}
• \begin{align*}a^{-m}=\frac{1}{a^m}\end{align*}
• \begin{align*}a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\end{align*}

You learned that scientific notation is a way to express large or small numbers in the form \begin{align*}x=a \times 10^n\end{align*} where \begin{align*}1 \le a < 10 \ \text{and} \ n \ \varepsilon \ I.\end{align*}

You learned to solve exponential equations with variables in the exponent you should try to rewrite the equations so the bases are the same. Then set the exponents equal to each other and solve. If the equation has a variable in the base you can try to get rid of the exponent or, make the exponents on each side of the equation the same and then set the bases equal to each other and solve.

Finally, you learned all about exponential functions. You learned for exponential functions of the form \begin{align*}y=ab^{\frac{x}{c}}+d\end{align*}, if \begin{align*}0 then the function is decreasing and represents exponential decay. If \begin{align*}b>1\end{align*} then the function is increasing and represents exponential growth. Exponential functions are used in many real-life situations such as with the decay of radioactive isotopes and with interest that compounds.

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