Power Functions and Exponents

In this document, we discuss power functions and how exponents affect them. 

 

Introduction

Power functions use rational numbers as exponents. There are several properties that define their behavior. We will go over these properties in some example problems.

 

Example 1

Simplify \( 16^3/4\)

\[\left( \frac{1}{t}\right)^{\!3/2}\]

You can look at this as saying the 4th root of 16 cubed.

\[ \sqrt[4]{16^3}\]

The fourth root of 16 = 2.

\[ (2)^3 = 8 \]

Example 2

Simplify: \( \left \frac{4^ \right {\!⅓}} {4^ \right {\!⅚}} \)

Put the numerator and denominator into one term and subtract the exponents.

\[= \left 4^{\right {\!⅓ - \!⅚} } \]

Subtract the exponents.

\[ = \left 4^{\right {\!-½}} \]

A negative exponent is just an inverse root. Let us move it to the denominator.

\[ = \frac{ 1}{\sqrt{4}} \]

Now, just simplify.

\[ = \frac{1}{2} \]