<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

11.2: Simplification of Radical Expressions

Difficulty Level: Basic Created by: CK-12
Atoms Practice
Estimated13 minsto complete
%
Progress
Practice Simplification of Radical Expressions
Practice
Progress
Estimated13 minsto complete
%
Practice Now
Turn In

Suppose that a shoemaker has determined that the optimal weight in ounces of a pair of running shoes is \begin{align*}\sqrt[4]{20000}\end{align*}200004. How many ounces would this be? Is there a way that you could rewrite this expression to make it easier to grasp? In this Concept, you'll learn how to simplify radical expressions like this one so that you can write them in multiple ways.

Guidance

Radicals are the roots of values. In fact, the word radical comes from the Latin word “radix,” meaning “root.” You are most comfortable with the square root symbol \begin{align*}\sqrt{x}\end{align*}x; however, there are many more radical symbols.

A radical is a mathematical expression involving a root by means of a radical sign.

\begin{align*}\sqrt[3]{y}=x && \text{because} \ x^3=y && \sqrt[3]{27}=3, \ because \ 3^3=27\\ \sqrt[4]{y}=x && \text{because} \ x^4=y && \sqrt[4]{16}=2 \ because \ 2^4=16\\ \sqrt[n]{y}=x && \text{because} \ x^n=y && \end{align*}y3=xy4=xyn=xbecause x3=ybecause x4=ybecause xn=y273=3, because 33=27164=2 because 24=16

Some roots do not have real values; in this case, they are called undefined.

Even roots of negative numbers are undefined.

\begin{align*}\sqrt[n]{x}\end{align*}xn is undefined when \begin{align*}n\end{align*}n is an even whole number and \begin{align*}x<0\end{align*}x<0.

Example A

Evaluate the following radicals:

  • \begin{align*}\sqrt[3]{64}\end{align*}643
  • \begin{align*}\sqrt[4]{-81}\end{align*}814

Solution:

\begin{align*}\sqrt[3]{64} = 4\end{align*}643=4 because \begin{align*}4^3=64\end{align*}43=64

\begin{align*}\sqrt[4]{-81}\end{align*}814 is undefined because \begin{align*}n\end{align*}n is an even whole number and \begin{align*}-81<0\end{align*}81<0.

In a previous Concept, you learned how to evaluate rational exponents:

\begin{align*}a^{\frac{x}{y}} \ where \ x=power \ and \ y=root\end{align*}axy where x=power and y=root

This can be written in radical notation using the following property.

Rational Exponent Property: For integer values of \begin{align*}x\end{align*}x and whole values of \begin{align*}y\end{align*}y:

\begin{align*}a^{\frac{x}{y}}= \sqrt[y]{a^x}\end{align*}axy=axy

Example B

Rewrite \begin{align*}x^{\frac{5}{6}}\end{align*}x56 using radical notation.

Solution:

This is correctly read as the sixth root of \begin{align*}x\end{align*}x to the fifth power. Writing in radical notation, \begin{align*}x^{\frac{5}{6}}=\sqrt[6]{x^5}\end{align*}x56=x56, where \begin{align*}x^5>0\end{align*}x5>0.

You can also simplify other radicals, like cube roots and fourth roots.

Example C

Simplify \begin{align*}\sqrt[3]{135}\end{align*}1353.

Solution:

Begin by finding the prime factorization of 135. This is easily done by using a factor tree.

\begin{align*}&\sqrt[3]{135}= \sqrt[3]{3 \cdot 3 \cdot 3 \cdot 5} = \sqrt[3]{3^3} \cdot \sqrt[3]{5}\\ & 3 \sqrt[3]{5}\end{align*}1353=33353=33353353

Video Review

Guided Practice

Evaluate \begin{align*}\sqrt[4]{4^2}\end{align*}424.

Solution: This is read, “The fourth root of four to the second power.”

\begin{align*}4^2=16\end{align*}42=16

The fourth root of 16 is 2; therefore,

\begin{align*}\sqrt[4]{4^2}=2\end{align*}424=2

Practice

Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Radical Expressions with Higher Roots (8:46)

  1. For which values of \begin{align*}n\end{align*}n is \begin{align*}\sqrt[n]{-16}\end{align*}16n undefined?

Evaluate each radical expression.

  1. \begin{align*}\sqrt{169}\end{align*}
  2. \begin{align*}\sqrt[4]{81}\end{align*}
  3. \begin{align*}\sqrt[3]{-125}\end{align*}
  4. \begin{align*}\sqrt[5]{1024}\end{align*}

Write each expression as a rational exponent.

  1. \begin{align*}\sqrt[3]{14}\end{align*}
  2. \begin{align*}\sqrt[4]{zw}\end{align*}
  3. \begin{align*}\sqrt{a}\end{align*}
  4. \begin{align*}\sqrt[9]{y^3}\end{align*}

Write the following expressions in simplest radical form.

  1. \begin{align*}\sqrt{24}\end{align*}
  2. \begin{align*}\sqrt{300}\end{align*}
  3. \begin{align*}\sqrt[5]{96}\end{align*}
  4. \begin{align*}\sqrt{\frac{240}{567}}\end{align*}
  5. \begin{align*}\sqrt[3]{500}\end{align*}
  6. \begin{align*}\sqrt[6]{64x^8}\end{align*}

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Vocabulary

radical

A mathematical expression involving a root by means of a radical sign. The word radical comes from the Latin word radix, meaning root.

Rational Exponent Property

For integer values of x and whole values of y: a^{\frac{x}{y}}= \sqrt[y]{a^x}

Radical Expression

A radical expression is an expression with numbers, operations and radicals in it.

Rationalize the denominator

To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

Image Attributions

Show Hide Details
Description
Difficulty Level:
Basic
Grades:
8 , 9
Date Created:
Feb 24, 2012
Last Modified:
Apr 28, 2016
Files can only be attached to the latest version of Modality
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.831.L.1
Here