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## Evaluate and estimate numerical square and cube roots

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The volume of a cube is equal to the measure of any one side to the power of three. This is written mathematically as \begin{align*}V=s^3.\end{align*} How can you determine the measure of any one side of the cube?

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*}; however, there are many more radical symbols.

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

\begin{align*}\text{If }x^3 & = y, \text{ then }\sqrt[3]{y}=x\\ \text{If }x^4 & = y, \text{ then }\sqrt[4]{y}=x\\ \text{If }x^n & = y, \text{ then }\sqrt[n]{y}=x\\ &\text{Similarly: }\\ \sqrt[3]{27} & =3, \text{ because } 3^3=27\\ \sqrt[4]{16}& =2 \text{ because } 2^4=16\end{align*}

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

Even-numbered roots of negative numbers are an example, since any number multiplied by itself an even number of times will produce a negative answer:

\begin{align*}(-1)(-1) & = +1\\ (-1)(-1)(-1)(-1) & = +1\\ (-1)(-1)(-1)(-1)(-1)(-1) &= +1\\ (-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1) &=+1\\ &\text{...and so on!} \end{align*}

This leads us to the general statement: "\begin{align*}\sqrt[n]{x}\end{align*} is undefined when \begin{align*}n\end{align*} is an even whole number and \begin{align*}x<0\end{align*}."

#### Let's evaluate the following radicals:

1. \begin{align*}\sqrt[3]{64}\end{align*}

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

1. \begin{align*}\sqrt[4]{-81}\end{align*}

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

#### Evaluating Rational Exponents

You may recall 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*} and whole values of \begin{align*}y\end{align*}:

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

#### Let's rewrite \begin{align*}x^{\frac{5}{6}}\end{align*} using radical notation:

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

#### Now, let's simplify the following radical:

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

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*}

### Examples

#### Example 1

Earlier, you were told that the volume of a cube is equal to the measure of any one side to the power of three. How can you determine the measure of any one side of the cube?

The statement "The volume of a cube is equal to the measure of any one side to the power of three" is written mathematically as \begin{align*}V=s^3.\end{align*}  That means you can determine the measure of any one side of a cube having given volume \begin{align*}V\end{align*} using \begin{align*}s=\sqrt[3]{V}.\end{align*} If the volume were 64, the side length would be 4 because \begin{align*}\sqrt[3]{64}=4\end{align*}

#### Example 2

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

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 and so:

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

### Review

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

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*}

To see the Review answers, open this PDF file and look for section 11.2.

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### Vocabulary Language: English Spanish

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 $y$: $a^{\frac{x}{y}}= \sqrt[y]{a^x}$

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.