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# 7.2: Rational Exponents and Roots

Difficulty Level: At Grade Created by: CK-12
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Practice Fractional Exponents

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A planet's maximum distance from the sun (in astronomical units) is given by the formula \begin{align*}d = p^{\frac {2}{3}}\end{align*}, were p is the period (in years) of the planet's orbit around the sun. If a planet's orbit around the sun is 27 years, what is its distance from the sun?

### The Rational Exponent Theorem

Now that you are familiar with nth roots, we will convert them into exponents. Let’s look at the square root and see if we can use the properties of exponents to determine what exponential number it is equivalent to.

#### Writing the Square Root as an Exponent

Step 1: Evaluate \begin{align*}\left(\sqrt{x}\right)^2\end{align*}. What happens?

The \begin{align*}\sqrt{\;\;}\end{align*} and the \begin{align*}^2\end{align*} cancel each other out, \begin{align*}\left(\sqrt{x}^2\right)=x\end{align*}.

Step 2: Recall that when a power is raised to another power, we multiply the exponents. Therefore, we can rewrite the exponents and root as an equation, \begin{align*}n\cdot 2=1\end{align*}. Solve for \begin{align*}n\end{align*}.

\begin{align*}\frac{n \cdot \cancel{2}}{\cancel{2}}&=\frac{1}{2} \\ n&=\frac{1}{2}\end{align*}

Step 3: From Step 2, we can conclude that \begin{align*}\sqrt{\;\;}=\frac{1}{2}\end{align*}.

\begin{align*}\left(\sqrt{x}\right)^2=\left(x^{\frac{1}{2}}\right)^2=x^{\left(\frac{1}{2}\right)\cdot 2}=x^1=x\end{align*}

From the steps above, we see that \begin{align*}\sqrt{x}=x^{\frac{1}{2}}\end{align*}. We can extend this idea to the other roots as well; \begin{align*}\sqrt[3]{x}=x^{\frac{1}{3}}=\sqrt[4]{x}=x^{\frac{1}{4}},\ldots \sqrt[n]{x}=x^{\frac{1}{n}}\end{align*}.

The Rational Exponent Theorem: For any real number \begin{align*}a\end{align*}, root \begin{align*}n\end{align*}, and exponent \begin{align*}m\end{align*}, the following is always true: \begin{align*}a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\end{align*}.

Let's solve the following problems.

1. Find \begin{align*}256^{\frac{1}{4}}\end{align*}.

Rewrite this expression in terms of roots. A number to the one-fourth power is the same as the fourth root.

\begin{align*}256^{\frac{1}{4}}=\sqrt[4]{256}=\sqrt[4]{4^4}=4\end{align*}

Therefore, \begin{align*}256^{\frac{1}{4}}=4\end{align*}.

1. Find \begin{align*}49^{\frac{3}{2}}\end{align*}.

This problem is similar to ones you have seen already. However, now, the root is written in the exponent. Rewrite the problem.

\begin{align*}49^{\frac{3}{2}}=\left(49^3\right)^{\frac{1}{2}}=\sqrt{49^3}\end{align*} or \begin{align*}\left(\sqrt{49}\right)^3\end{align*}

It is easier to evaluate the second option above.

So \begin{align*}\left(\sqrt{49}\right)^3=7^3=343\end{align*}.

1. Find \begin{align*}5^{\frac{2}{3}}\end{align*} using a calculator. Round your answer to the nearest hundredth.

To type this into a calculator, the keystrokes would probably look like: \begin{align*}5^{\frac{2}{3}}\end{align*}. The “^” symbol is used to indicate a power. Anything in parenthesis after the “^” would be in the exponent. Evaluating this, we have 2.924017738..., or just 2.92.

Other calculators might have a \begin{align*}x^y\end{align*} button. This button has the same purpose as the ^ and would be used in the exact same way.

### Examples

#### Example 1

Earlier, you were asked to find the planet's distance from the sun.

Substitute 27 for p and solve.

\begin{align*}d = 27^{\frac{2}{3}}\end{align*}

Rewrite the problem.

\begin{align*}27^{\frac{2}{3}}=\left(27^2\right)^{\frac{1}{3}}=\sqrt[3]{27^2}\end{align*} or \begin{align*}\sqrt[3]{27}^2\end{align*}

\begin{align*}\left(\sqrt[3]{27}\right)^2=3^2=9\end{align*}.

Therefore, the planet's distance from the sun is 9 astronomical units.

#### Example 2

Rewrite \begin{align*}\sqrt[7]{12}\end{align*} using rational exponents. Then, use a calculator to find the answer.

Using rational exponents, the \begin{align*}7^{th}\end{align*} root becomes the \begin{align*}\frac{1}{7}\end{align*} power;\begin{align*}12^{\frac{1}{7}}=1.426\end{align*}.

#### Example 3

Rewrite \begin{align*}845^{\frac{4}{9}}\end{align*} using roots. Then, use a calculator to find the answer.

Using roots, the 9 in the denominator of the exponent is the root;\begin{align*}\sqrt[9]{845^4}=19.99\end{align*}. To enter this into a calculator, you can use the rational exponents. If you have a TI-83 or 84, press MATH and select 5: \begin{align*}\sqrt[x]{\;\;}\end{align*}. On the screen, you should type \begin{align*}9\sqrt[x]{\;\;} \ 845^\land 4\end{align*} to get the correct answer. You can also enter \begin{align*}845^\land {\left(\frac{4}{9}\right)}\end{align*} and get the exact same answer

#### Example 4

Evaluate without a calculator: \begin{align*}125^{\frac{4}{3}}\end{align*}.

\begin{align*} 125^{\frac{4}{3}}=\left(\sqrt[3]{125}\right)^4=5^4=625\end{align*}

#### Example 5

Evaluate without a calculator: \begin{align*}256^{\frac{5}{8}}\end{align*}.

\begin{align*}256^{\frac{5}{8}}=\left(\sqrt[8]{256}\right)^5=2^5=32\end{align*}

#### Example 6

Evaluate without a calculator: \begin{align*}\sqrt{81^{\frac{1}{2}}}\end{align*}.

### Review

Write the following expressions using rational exponents and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.

1. \begin{align*}\sqrt[5]{45}\end{align*}
2. \begin{align*}\sqrt[9]{140}\end{align*}
3. \begin{align*}\sqrt[8]{50}^3\end{align*}

Write the following expressions using roots and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.

1. \begin{align*}72^{\frac{5}{3}}\end{align*}
2. \begin{align*}95^{\frac{2}{3}}\end{align*}
3. \begin{align*}125^{\frac{3}{4}}\end{align*}

Evaluate the following without a calculator.

1. \begin{align*}64^{\frac{2}{3}}\end{align*}
2. \begin{align*}27^{\frac{4}{3}}\end{align*}
3. \begin{align*}16^{\frac{5}{4}}\end{align*}
4. \begin{align*}\sqrt{25^3}\end{align*}
5. \begin{align*}\sqrt[2]{9}^5\end{align*}
6. \begin{align*}\sqrt[5]{32^2}\end{align*}

For the following problems, rewrite the expressions with rational exponents and then simplify the exponent and evaluate without a calculator.

1. \begin{align*}\sqrt[4]{\left(\frac{2}{3}\right)^8}\end{align*}
2. \begin{align*}\sqrt[3]{\frac{7}{2}}^6\end{align*}
3. \begin{align*}\sqrt{\left(16\right)^{\frac{1}{2}}}^6\end{align*}

To see the Review answers, open this PDF file and look for section 7.2.

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### Vocabulary Language: English

TermDefinition
Rational Exponent A rational exponent is a fractional exponent.

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