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# Fractional Exponents

## Relate fractional exponents to nth roots

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Fractional Exponents

Suppose you want to find the cube root of a number, but your calculator only allows you to find square roots. However, it does allow you to find a number raised to any power, including non-integer powers. What would you do? Is there a way that you could rewrite the cube root of a number with a fractional exponent so that you could use your calculator to find the answer?

### Fractional Exponents

The next objective is to be able to use fractions as exponents in an expression.

You can write roots as fractional exponents using the rule \begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*}.

#### Let's simplify the following expressions by writing them without a root:

1.  \begin{align*}\sqrt{a}\\a^{\frac{1}{2}} \end{align*}
2.  \begin{align*}\sqrt[3]{a}\\a^{\frac{1}{3}}\end{align*}
3.  \begin{align*}\sqrt[5]{a^2}\\(a^2)^{\frac{1}{5}}\\a^{\frac{2}{5}}\end{align*}
4. \begin{align*}\sqrt[3]{\chi}\end{align*}

\begin{align*}\chi^{\frac{1}{3}}\end{align*}

1.  \begin{align*}\sqrt[4]{\chi^3}\end{align*}

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

#### Evaluating Expressions with Exponents

It is important when evaluating expressions that you remember the Order of Operations. Evaluate what is inside the parentheses, then evaluate the exponents, then perform multiplication/division from left to right, and then perform addition/subtraction from left to right.

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

\begin{align*}3 \cdot 5^2 - 9^{1/2} \cdot 5+1\end{align*}

\begin{align*}3 \cdot 5^2-9^{1/2} \cdot 5+1=3 \cdot 25-\sqrt{9} \cdot 5+1=75-3\cdot 50+1=75-150+1=-74\end{align*}

### Examples

#### Example 1

Earlier, you were asked how you can write the cube root so that you can use your calculator to find cube root of a number.

As shown in this concept, you can write a root as a fractional exponent where the denominator is the power of the root and the numerator is the power that is inside the root.

A cube root can be written as the number to the \begin{align*}\frac{1}{3}\end{align*} power.

\begin{align*}\sqrt[3]{x}= x^{\frac{1}{3}}\end{align*}

#### Example 2

Simplify \begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}\end{align*}.

We start by simplifying using the fact that \begin{align*}36^{3/2}=(36^{1/2})^3\end{align*}.

\begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (36^{1/2})^3}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (\sqrt{36})^3}{2^3}\right)^{2/5} = \left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}\end{align*}

Next we rearrange knowing that 6 and 2 have a common factor, 2.

\begin{align*}\left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}=\left( 3^2 \cdot \left(\frac{6}{2}\right)^3\right)^{2/5} = \left( 3^2 \cdot 3^3\right)^{2/5} =\left(3^5\right)^{2/5} =3^{5\cdot 2/5}=3^2=9\end{align*}

### Review

Simplify the following expressions. Be sure the final answer includes only positive exponents.

1. \begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}
2. \begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{align*}
3. \begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}
4. \begin{align*}\frac{x^{-3}y^{-5}}{z^{-7}}\end{align*}
5. \begin{align*}(x^{\frac{1}{2}} y^{-\frac{2}{3}})(x^2 y^{\frac{1}{3}})\end{align*}

Simplify the following expressions without any fractions in the answer.

1. \begin{align*}\left(\frac{3x}{y^{\frac{1}{3}}}\right)^3\end{align*}
2. \begin{align*}\frac{4a^2b^3}{2a^5b}\end{align*}
3. \begin{align*}\left(\frac{x}{3y^2}\right)^3 \cdot \frac{x^2y}{4}\end{align*}
4. \begin{align*}\left(\frac{ab^{-2}}{b^3}\right)^2\end{align*}
5. \begin{align*}\frac{x^{-3}y^2}{x^2y^{-2}}\end{align*}
6. \begin{align*}\frac{3x^2y^{\frac{3}{2}}}{xy^{\frac{1}{2}}}\end{align*}
7. \begin{align*}\frac{(3x^3)(4x^4)}{(2y)^2}\end{align*}
8. \begin{align*}\frac{a^{-2}b^{-3}}{c^{-1}}\end{align*}
9. \begin{align*}\frac{x^{\frac{1}{2}}y^{\frac{5}{2}}}{x^{\frac{3}{2}}y^{\frac{3}{2}}}\end{align*}

Evaluate the following expressions to a single number.

1. \begin{align*}(16^{\frac{1}{2}})^3\end{align*}
2. \begin{align*}5^0\end{align*}
3. \begin{align*}7^2\end{align*}
4. \begin{align*}\left(\frac{2}{3}\right)^3\end{align*}
5. \begin{align*}3^{-3}\end{align*}
6. \begin{align*}16^{\frac{1}{2}}\end{align*}
7. \begin{align*}8^{\frac{-1}{3}}\end{align*}

Mixed Review

1. A quiz has ten questions: 7 true/false and 3 multiple choice. The multiple choice questions each have four options. How many ways can the test be answered?
2. Simplify \begin{align*}3a^4 b^4 \cdot a^{-3} b^{-4}\end{align*}.
3. Simplify \begin{align*}(x^4 y^2 \cdot xy^0)^5\end{align*}.
4. Simplify \begin{align*}\frac{v^2}{-vu^{-2} \cdot u^{-1} v^4}\end{align*}.
5. Solve for \begin{align*}n: -6(4n+3)=n+32\end{align*}.

To view the Review answers, open this PDF file and look for section 8.4.

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TermDefinition
roots as fractional exponents $\sqrt[m]{a^n}=a^{\frac{n}{m}}$.

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