Roots and Radicals
Simplify Rational Exponents
Learning Objectives
By the end of this section, you will be able to:
- Simplify expressions with
- Simplify expressions with
- Use the properties of exponents to simplify expressions with rational exponents
Before you get started, take this readiness quiz.
Simplify Expressions with
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that We will use the Power Property of Exponents to find the value of p.
So But we know also Then it must be that
This same logic can be used for any positive integer exponent n to show that
If is a real number and then
The denominator of the rational exponent is the index of the radical.
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
Write as a radical expression: ⓐ ⓑ ⓒ
We want to write each expression in the form
ⓐ
ⓑ
ⓒ
Write as a radical expression: ⓐ ⓑ ⓒ
ⓐⓑⓒ
Write as a radial expression: ⓐ ⓑ ⓒ
ⓐⓑⓒ
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
Write with a rational exponent: ⓐ ⓑ ⓒ
We want to write each radical in the form
ⓐ
ⓑ
*** QuickLaTeX cannot compile formula: \begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\sqrt[3]{4x}\hfill \\ \begin{array}{c}\text{The index is 3, so the denominator of the}\hfill \\ \text{exponent is 3. Include parentheses}\phantom{\rule{0.2em}{0ex}}\left(4x\right).\hfill \end{array}\hfill & & & \hfill \begin{array}{}\\ \\ \hfill \phantom{\rule{5em}{0ex}}{\left(4x\right)}^{\frac{1}{3}}\hfill \end{array}\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: ...d{array}\hfill & & & \hfill \begin{array}{} Missing $ inserted. leading text: ...\ \\ \hfill \phantom{\rule{5em}{0ex}}{\left Extra }, or forgotten $. leading text: ... \phantom{\rule{5em}{0ex}}{\left(4x\right)} Missing } inserted. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array} Extra }, or forgotten $. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array} Missing } inserted. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array} Extra }, or forgotten $. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array} Missing } inserted. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array} Extra }, or forgotten $. leading text: ...(4x\right)}^{\frac{1}{3}}\hfill \end{array}
ⓒ
*** QuickLaTeX cannot compile formula: \begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}3\ph