# 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.

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