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# Operations with Rational Exponents

## Simplify expressions with negative and fractional exponents.

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Practice Operations with Rational Exponents

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Applying the Laws of Exponents to Rational Exponents

The period (in seconds) of a pendulum with a length of L (in meters) is given by the formula \begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}. If the length of a pendulum is \begin{align*}9.8^{\frac{8}{3}}\end{align*}, what is its period?

### Applying the Laws of Exponents to Rational Exponents

When simplifying expressions with rational exponents, all the laws of exponents that were learned previously are still valid. On top of that, the rules of fractions apply as well.

Let's simplify the following.

1. \begin{align*}x^{\frac{1}{2}} \cdot x^{\frac{3}{4}}\end{align*}

Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we add the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators.

\begin{align*}x^{\frac{1}{2}}\cdot x^{\frac{3}{4}}=x^{\frac{2}{4}}\cdot x^{\frac{3}{4}}=x^{\frac{5}{4}}\end{align*}

This rational exponent does not reduce, so we are done.

1. \begin{align*}\frac{4x^{\frac{2}{3}}y^4}{16x^3y^{\frac{5}{6}}}\end{align*}

This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and reduce \begin{align*}\frac{4}{16}\end{align*}.

\begin{align*}\frac{4x^{\frac{2}{3}}y^4}{16x^3y^{\frac{5}{6}}}=\frac{1}{4}x^{{\left(\frac{2}{3}\right)}-3}y^{\frac{4-5}{6}}=\frac{1}{4}x^{\frac{-7}{3}}y^{\frac{19}{6}}\end{align*}

If you are writing your answer in terms of positive exponents, your answer would be \begin{align*}\frac{y^{\frac{19}{6}}}{4x^{\frac{7}{3}}}\end{align*}. Notice, that when a rational exponent is improper we do not change it to a mixed number.

If we were to write the answer using roots, then we would take out the whole numbers. For example, \begin{align*}y= \frac{19}{6}\end{align*} can be written as \begin{align*}y^{\frac{19}{6}}=y^3y^{\frac{1}{6}}=y^3\sqrt[6]{y}\end{align*} because 6 goes into 19, 3 times with a remainder of 1.

1. \begin{align*}\frac{\left(2x^{\frac{1}{2}}y^6\right)^{\frac{2}{3}}}{4x^{\frac{5}{4}}y^{\frac{9}{4}}}\end{align*}

On the numerator, the entire expression is raised to the \begin{align*}\frac{2}{3}\end{align*} power. Distribute this power to everything inside the parenthesis. Then, use the Powers Property of Exponents and rewrite 4 as \begin{align*}2^2\end{align*}.

\begin{align*}\frac{\left(2x^{\frac{1}{2}}y^6\right)^{\frac{2}{3}}}{4x^{\frac{5}{4}}y^{\frac{9}{4}}}=\frac{2^{\frac{2}{3}}x^{\frac{1}{3}}y^4}{2^2x^{\frac{5}{4}}y^{\frac{9}{4}}}\end{align*}

Combine like terms by subtracting the exponents.

\begin{align*}\frac{2^{\frac{2}{3}}x^{\frac{1}{3}}y^4}{2^2x^{\frac{5}{4}}y^{\frac{9}{4}}} = 2^{\left(\frac{2}{3}\right)-2}x^{\left(\frac{1}{3}\right)-\left(\frac{5}{4}\right)}y^{4-\left(\frac{9}{4}\right)}=2^{\frac{-4}{3}}x^{\frac{-11}{12}}y^{\frac{7}{4}}\end{align*}

Finally, rewrite the answer with positive exponents by moving the 2 and \begin{align*}x\end{align*} into the denominator. \begin{align*}\frac{y^{\frac{7}{4}}}{2^{\frac{4}{3}}x^{\frac{11}{12}}}\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the period of the pendulum.

Substitute \begin{align*}9.8^{\frac{8}{3}}\end{align*} for L and solve.

\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\\ P = 2\pi{(\frac{9.8^{\frac{8}{3}}}{9.8})}^{\frac{1}{2}}\\ P = 2\pi{(\frac{9.8^{\frac{8}{3}}}{9.8^{\frac{3}{3}}})^{\frac{1}{2}}}\\ P = 2\pi{(9.8^{\frac{5}{3}})^{\frac{1}{2}}}\\ P = 2\pi{(9.8)^{\frac{5}{6}}}\end{align*}

Therefore, the period of the pendulum is \begin{align*} P = 2\pi{(9.8)^{\frac{5}{6}}}\end{align*}.

Simplify each expression below. Reduce all rational exponents and write final answers using positive exponents.

#### Example 2

\begin{align*}4d^{\frac{3}{5}} \cdot 8^{\frac{1}{3}}d^{\frac{2}{5}}\end{align*}

Change 4 and 8 so that they are powers of 2 and then add exponents with the same base.

\begin{align*}4d^{\frac{3}{5}} \cdot 8^{\frac{1}{3}}d^{\frac{2}{5}}=2^2 d^{\frac{3}{5}} \cdot \left(2^3\right)^{\frac{1}{3}}d^{\frac{2}{5}}=2^3 d^{\frac{5}{5}}=8d\end{align*}

#### Example 3

\begin{align*}\frac{w^{\frac{7}{4}}}{w^{\frac{1}{2}}}\end{align*}

Subtract the exponents. Change the \begin{align*}\frac{1}{2}\end{align*} power to \begin{align*}\frac{2}{4}\end{align*}.

\begin{align*}\frac{w^{\frac{7}{4}}}{w^\frac{1}{2}}= \frac{w^{\frac{7}{4}}}{w^{\frac{2}{4}}}=w^{\frac{5}{4}}\end{align*}

#### Example 4

\begin{align*}\left(3^{\frac{3}{2}}x^4 y^{\frac{6}{5}}\right)^{\frac{4}{3}}\end{align*}

Distribute the \begin{align*}\frac{4}{3}\end{align*} power to everything inside the parenthesis and reduce.

\begin{align*}\left(3^{\frac{3}{2}}x^4 y^{\frac{6}{5}}\right)^{\frac{4}{3}}=3^{\frac{12}{6}}x^{\frac{16}{3}}y^{\frac{24}{15}}=3^2 x^{\frac{16}{3}}y^{\frac{8}{5}}=9x^{\frac{16}{3}}y^{\frac{8}{5}}\end{align*}

### Review

Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.

1. \begin{align*}\frac{1}{5}a^{\frac{4}{5}}25^{\frac{3}{2}}a^{\frac{3}{5}}\end{align*}
2. \begin{align*}7b^{\frac{4}{3}}49^{\frac{1}{2}}b^{-\frac{2}{3}}\end{align*}
3. \begin{align*}\frac{m^{\frac{8}{9}}}{m^{\frac{2}{3}}}\end{align*}
4. \begin{align*}\frac{x^{\frac{4}{7}}y^{\frac{11}{6}}}{x^{\frac{1}{14}}y^{\frac{5}{3}}}\end{align*}
5. \begin{align*}\frac{8^{\frac{5}{3}}r^5 s^{\frac{3}{4}}t^{\frac{1}{3}}}{2^4 r^{\frac{21}{5}}s^2 t^{\frac{7}{9}}}\end{align*}
6. \begin{align*}\left(a^{\frac{3}{2}}b^{\frac{4}{5}}\right)^{\frac{10}{3}}\end{align*}
7. \begin{align*}\left(5x^{\frac{5}{7}}y^4\right)^{\frac{3}{2}}\end{align*}
8. \begin{align*}\left(\frac{4x^{\frac{2}{5}}}{9y^{\frac{4}{5}}}\right)^{\frac{5}{2}}\end{align*}
9. \begin{align*}\left(\frac{75d^{\frac{18}{5}}}{3d^{\frac{3}{5}}}\right)^{\frac{5}{2}}\end{align*}
10. \begin{align*}\left(\frac{81^{\frac{3}{2}}a^3}{8a^{\frac{9}{2}}}\right)^{\frac{1}{3}}\end{align*}
11. \begin{align*}27^{\frac{2}{3}}m^{\frac{4}{5}}n^{-\frac{3}{2}}4^{\frac{1}{2}}m^{-\frac{2}{3}}n^{\frac{8}{5}}\end{align*}
12. \begin{align*}\left(\frac{3x^{\frac{3}{8}}y^{\frac{2}{5}}}{5x^{\frac{1}{4}}y^{-\frac{3}{10}}}\right)^2\end{align*}

To see the Review answers, open this PDF file and look for section 7.3.

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