11.2: Simplification of Radical Expressions
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*}
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*}
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*}
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*}
Example A
Evaluate the following radicals:

\begin{align*}\sqrt[3]{64}\end{align*}
64−−√3 
\begin{align*}\sqrt[4]{81}\end{align*}
−81−−−−√4
Solution:
\begin{align*}\sqrt[3]{64} = 4\end{align*}
\begin{align*}\sqrt[4]{81}\end{align*}
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*}
This can be written in radical notation using the following property.
Rational Exponent Property: For integer values of \begin{align*}x\end{align*}
\begin{align*}a^{\frac{x}{y}}= \sqrt[y]{a^x}\end{align*}
Example B
Rewrite \begin{align*}x^{\frac{5}{6}}\end{align*}
Solution:
This is correctly read as the sixth root of \begin{align*}x\end{align*}
You can also simplify other radicals, like cube roots and fourth roots.
Example C
Simplify \begin{align*}\sqrt[3]{135}\end{align*}
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*}
Guided Practice
Evaluate \begin{align*}\sqrt[4]{4^2}\end{align*}
Solution: This is read, “The fourth root of four to the second power.”
\begin{align*}4^2=16\end{align*}
The fourth root of 16 is 2; therefore,
\begin{align*}\sqrt[4]{4^2}=2\end{align*}
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. CK12 Basic Algebra: Radical Expressions with Higher Roots (8:46)
 For which values of \begin{align*}n\end{align*}
n is \begin{align*}\sqrt[n]{16}\end{align*}−16−−−−√n undefined?
Evaluate each radical expression.

\begin{align*}\sqrt{169}\end{align*}
169−−−√ 
\begin{align*}\sqrt[4]{81}\end{align*}
81−−√4 
\begin{align*}\sqrt[3]{125}\end{align*}
−125−−−−√3 
\begin{align*}\sqrt[5]{1024}\end{align*}
1024−−−−√5
Write each expression as a rational exponent.

\begin{align*}\sqrt[3]{14}\end{align*}
14−−√3 
\begin{align*}\sqrt[4]{zw}\end{align*}
zw−−−√4 
\begin{align*}\sqrt{a}\end{align*}
a√  \begin{align*}\sqrt[9]{y^3}\end{align*}
Write the following expressions in simplest radical form.
 \begin{align*}\sqrt{24}\end{align*}
 \begin{align*}\sqrt{300}\end{align*}
 \begin{align*}\sqrt[5]{96}\end{align*}
 \begin{align*}\sqrt{\frac{240}{567}}\end{align*}
 \begin{align*}\sqrt[3]{500}\end{align*}
 \begin{align*}\sqrt[6]{64x^8}\end{align*}
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Term  Definition 

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 and whole values of : 
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
Here you'll learn how to simplify radical expressions.