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Fractional Exponents

Relate fractional exponents to nth roots

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Rational Exponents and Roots

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?

Guidance

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.

Investigation: Writing the Square Root as an Exponent

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, @$\begin{align*}\left(\sqrt{x}^2\right)=x\end{align*}@$ .

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 @$\begin{align*}n\end{align*}@$ .

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

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

Example A

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

Solution: 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, @$\begin{align*}256^{\frac{1}{4}}=4\end{align*}@$ .

Example B

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

Solution: This problem is the same as the ones in the previous concept. 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 @$\begin{align*}\left(\sqrt{49}\right)^3\end{align*}@$

From the previous concept, we know that it is easier to evaluate the second option above. @$\begin{align*}\left(\sqrt{49}\right)^3=7^3=343\end{align*}@$ .

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: @$\begin{align*}a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\end{align*}@$ .

Example C

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

Solution: 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.

Intro Problem Revisit 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.

Guided Practice

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

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

Evaluate without a calculator.

3. @$\begin{align*}125^{\frac{4}{3}}\end{align*}@$

4. @$\begin{align*}256^{\frac{5}{8}}\end{align*}@$

5. @$\begin{align*}\sqrt{81^{\frac{1}{2}}}\end{align*}@$

Answers

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

2. 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

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

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

5. @$\begin{align*}\sqrt{81^{\frac{1}{2}}}=\sqrt{\sqrt{81}}=\sqrt{9}=3\end{align*}@$

Explore More

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. @$\begin{align*}\sqrt{\left(16\right)^{\frac{1}{2}}}^6\end{align*}@$

Vocabulary

Rational Exponent

Rational Exponent

A rational exponent is a fractional exponent.

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