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# Fractional Exponents

## Relate fractional exponents to nth roots

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Evaluate Radical Expressions and Fractional Powers

The first dance of the year at the local middle school was held on the last Friday in September. The school council decided that the price of admission into the dance would be a non-perishable food item which would be donated to the local food bank. On Monday, the members of the students’ council sorted the food items into dry goods and canned goods.

The total number of canned goods was 321. The principle asked the students to represent 321 using a mathematical expression and the writer of the best expression would win free admission into the next dance. How does this winning expression represent 321 cans?

In this concept, you will learn to evaluate radical expressions and fractional exponents.

Remember a radical consists of a radical sign, a radicand and an index. The radical sign is the symbol you use to denote square root, \begin{align*}\sqrt{\;\;}\end{align*}, the radicand is the number under the radical sign, \begin{align*}\sqrt{\text{radicand}}\end{align*}, and the index is the number that tells what root to find, \begin{align*}\sqrt[{\color{red}3}]{\text{radicand}}\end{align*}. The cube root of 27 can be expressed as \begin{align*}\sqrt[3]{27}\end{align*}.

In the radical expression \begin{align*}\sqrt[3]{27}\end{align*} the exponent of 27 is understood as being one. Therefore the radical expression could also be written as \begin{align*}\sqrt[3]{27^1}\end{align*} or as \begin{align*}\left(\sqrt[3]{27} \right)^1\end{align*}. The number that raises a base to a power is called an exponent. All of the exponents that you have used so far have been whole numbers. However, an exponent can also be a fraction such that its numerator is the exponent of the base and its denominator is the index of the radical. The base of a radical expression is the radicand. This means that the radical expression \begin{align*}\sqrt[3]{27}\end{align*} can also be written using a fractional exponent as \begin{align*}27^\frac{\text{exponent}}{\text{index}}\end{align*} or \begin{align*}27^\frac{1}{3}\end{align*}.

The square root of 16 can written as the radical expression \begin{align*}\sqrt{16}\end{align*}. The index ‘2’ is not written here but is understood. The exponent of ‘1’ for 16 is also not written but is understood. Therefore the \begin{align*}\sqrt{16}\end{align*} can be written as \begin{align*}16^\frac{1}{2}\end{align*} using fractional exponents.

Whether the expression is written in radical form or fractional form it can be evaluated.

\begin{align*}27^{\frac{1}{3}} \ \text{or} \ \sqrt[3]{27}\end{align*}

Both of these expressions mean to find the cube root of 27. The cube of 27 is the number which multiplied by itself three times equals 27.

\begin{align*}27 = 3 \times 3 \times 3 = 3^3\end{align*}

The cube root of 27 is 3.

### Examples

#### Example 1

Earlier, you were given a problem about the dance and all the canned goods. You need to figure out how \begin{align*}361^\frac{1}{2} + 302\end{align*} equals 321 cans of food.

First, write the fractional power as a radical expression.

\begin{align*}\sqrt{361}\end{align*}

Next, write the entire expression.

\begin{align*}\sqrt{361} + 302\end{align*}

Next, determine the square of 361.

\begin{align*}361 = 19 \times 19 = 19^2\end{align*}. The square is 19.

Next, substitute 19 into the expression for \begin{align*}\sqrt{361}\end{align*}.

\begin{align*}19 + 302\end{align*}

\begin{align*}19 + 302 = 321\end{align*}

The expression represents 321 cans of food.

#### Example 2

Evaluate the following fractional power.

\begin{align*}64^\frac{2}{3}\end{align*}

First, determine what the fractional represents. The ‘2’ is the exponent and the ‘3’ is the index.

Next, rewrite the fractional power as a radical. Remember, the exponent \begin{align*}\frac{2}{3}\end{align*} represents \begin{align*}\frac{\text{exponent}}{\text{index}}\end{align*}.

\begin{align*}\left(\sqrt[3]{64} \right)^2\end{align*}

Next, determine the cube of 64. It is the number which multiplied by itself three times equals 64.

\begin{align*}64 = 4 \times 4 \times 4 = 4^3\end{align*}. The cube of 64 is 4.

Next, write the new expression by substituting 4 for \begin{align*}\sqrt[3]{64}\end{align*}.

\begin{align*}(4)^2\end{align*}

Then, evaluate the new expression.

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

#### Example 3

Evaluate each of the following expressions.

1. \begin{align*}\sqrt{121}\end{align*}
2. \begin{align*}\sqrt[4]{16}\end{align*}
3. \begin{align*}125^\frac{2}{3}\end{align*}

a) \begin{align*}\sqrt{121}\end{align*}

First, determine what the expression means.

Find the square root of 121. The square root is the number which multiplied by itself twice equals 121.

Next, determine the square of 121.

\begin{align*}121 = 11 \times 11 = 11^2\end{align*}

Then, write the expression equal to its square.

\begin{align*}\sqrt{121} = 11\end{align*}

b) \begin{align*}\sqrt[4]{16}\end{align*}

First, determine what the expression means.

Find the fourth root of 121. The fourth root is the number which multiplied by itself four times equals 16.

Next, determine the fourth root of 16.

\begin{align*}\begin{array}{rcl} 16 &=& 2 \times 2 = {\color{blue}4} \times 2 = {\color{blue}8} \times 2 = {\color{blue}16}\\ 16 &=& 2 \times 2 \times 2 \times 2 = 2^4 \end{array}\end{align*}

Then, write the expression equal to its fourth root.

\begin{align*}\sqrt[4]{16} = 2\end{align*}

c) \begin{align*}125^\frac{2}{3}\end{align*}

First, determine what the expression means.

Find the cube root of 125 and square the cube root.

Next, write the expression to represent its meaning.

\begin{align*}\left ( 125^\frac{1}{3} \right )^2\end{align*}

Next, perform the operation in the parenthesis. Determine the cube root of 125.

\begin{align*}\begin{array}{rcl} 125 &=& 5 \times 5 = {\color{blue}25} \times 5 = {\color{blue}125}\\ 125 &=& 5 \times 5 \times 5 = 5^3 \end{array}\end{align*}

Next, write the new expression by substituting ‘5’ for \begin{align*}\left ( 125^\frac{1}{3} \right ) \end{align*}.

\begin{align*}(5)^2\end{align*}

Next, evaluate the new expression.

\begin{align*}(5)^2 = 5 \times 5 = 25\end{align*}

Then, write the given expression equal to the evaluated value.

\begin{align*}\left ( 125^\frac{1}{3} \right )^2 = 25\end{align*}

#### Example 4

Rewrite the following radical expression using fractional exponents.

\begin{align*}\left ( \sqrt[4]{81} \right )^3\end{align*}

First, recall what is represented by a fractional exponent.

The numerator represents the exponent and the denominator represents the index. \begin{align*}\left(\frac{\text{exponent}}{\text{index}}\right)\end{align*}

\begin{align*}\left (\sqrt[\text{index } \rightarrow \ 4]{81} \right )^{3 \ \leftarrow \text{ exponent}}\end{align*}

Next, write the base (radicand) with the fractional exponent.

\begin{align*}81^\frac{3}{4}\end{align*}

Then, write the radical expression equal to the fractional power.

\begin{align*}(\sqrt[4]{81})^3 = 81^\frac{3}{4}\end{align*}

The answer is \begin{align*}81^\frac{3}{4}\end{align*}.

#### Example 5

Rewrite the following expression as a radical.

\begin{align*}32^\frac{3}{5}\end{align*}

First, recall what the fractional exponent represents.

The fractional exponent represents \begin{align*}\left(\frac{\text{exponent}}{\text{index}}\right)\end{align*}.

Next, write the base of 32 as the radicand of a radical expression.

\begin{align*}\sqrt{32}\end{align*}

Next, write the index in its correct position to indicate what root of 32 needs to be found.

\begin{align*}\sqrt[5]{32}\end{align*}

Next, write the exponent in its correct position to indicate that the fifth root of 32 must be raised to the power of 3.

\begin{align*}\left ( \sqrt[5]{32} \right )^3\end{align*}

Then, write the fractional power equal to the radical expression.

\begin{align*}32^\frac{3}{5} = \left ( \sqrt[5]{32} \right )^3\end{align*}

The answer is \begin{align*}\left ( \sqrt[5]{32} \right )^3\end{align*}.

### Review

Evaluate each fractional power.

1.  \begin{align*}64^\frac{1}{2}\end{align*}
2. \begin{align*}16^\frac{1}{2}\end{align*}
3. \begin{align*}144^\frac{1}{2}\end{align*}
4. \begin{align*}81^\frac{1}{2}\end{align*}
5. \begin{align*}9^\frac{1}{2}\end{align*}
6. \begin{align*}25^\frac{1}{2}\end{align*}
7. \begin{align*}216^\frac{1}{3}\end{align*}
8. \begin{align*}100^\frac{1}{2}\end{align*}
9. \begin{align*}16^\frac{1}{4}\end{align*}
10. \begin{align*}256^\frac{1}{4}\end{align*}
11. \begin{align*}125^\frac{1}{3}\end{align*}
12. \begin{align*}36^\frac{1}{2}\end{align*}
13. \begin{align*}81^\frac{1}{4}\end{align*}
14. \begin{align*}121^\frac{1}{2}\end{align*}
15. \begin{align*}169^\frac{1}{2}\end{align*}

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