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College Algebra : Rational Exponents

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Example Question #10 : Fractional Exponents

Simplify: $25^{\frac{3}{2}}$

Possible Answers:

125

Correct answer:

125

Explanation:

An option to solve this is to split up the fraction. Rewrite the fractional exponent as follows:

$25^{\frac{3}{2}} = (25^{\frac{1}{2}})^3$

A value to its half power is the square root of that value.

$25^{\frac{1}{2}} = \sqrt{25}=5$

Substitute this value back into $(25^{\frac{1}{2}})^3$ .

$(25^{\frac{1}{2}})^3 = (5)^3 = 5\times 5\times 5 = 125$

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Example Question #1 : Rational Exponents

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

Possible Answers:

$\sqrt[4]{b^5}$

$\sqrt[5]{b^4}$

$\sqrt[4]{b^{-5}}$

5b^4

Correct answer:

$\sqrt[5]{b^4}$

Explanation:

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

When dealing with fractional exponents, keep the following in mind: The numerator is making the base bigger, so treat it like a regular exponent. The denominator is making the base smaller, so it must be the root you are taking.

This means that $b^{\frac{4}{5}}$ is equal to the fifth root of b to the fourth. Perhaps a bit confusing, but it means that we will keep b^4 , but put the whole thing under $\sqrt[5]{b}$ .

So if we put it together we get:

$\sqrt[5]{b^4}$

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Example Question #11 : Fractional Exponents

Evaluate $\frac{1}{4}^{-\frac{1}{2}}$

Possible Answers:

$-\frac{1}{16}$

$\frac{1}{4}$

$\frac{1}{16}$

Correct answer:

Explanation:

When dealing with fractional exponents, remember this form:

$\sqrt[a]{x^n}$ is the index of the radical which is also the denominator of the fraction, represents the base of the exponent, and is the power the base is raised to. That value is the numerator of the fraction.

With a negative exponent, we need to remember this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{4}^{-\frac{1}{2}}=\frac{1}{\frac{1}{4}^{\frac{1}{2}}}=\frac{1}{\sqrt[2]{\frac{1}{4^1}}}=\frac{1}{\frac{1}{2}}=2$

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Example Question #3 : Rational Exponents

$\left ( \frac{125}{343} \right )^{\frac{2}{3}}$

Possible Answers:

$\frac{5}{7}$

$\frac{25}{49}$

$\frac{5}{343}$

$\frac{625}{2401}$

$\frac{250}{686}$

Correct answer:

$\frac{25}{49}$

Explanation:

First, distribute the exponent to both the numerator and denominator of the fraction.

$\frac{125^{\frac{2}{3}}}{343^{\frac{2}{3}}}$

The numerator of a fractional exponent is the power you take the number to and the denominator is the root that you take the number to.

$\frac{\sqrt[3]{125^{2}}}{\sqrt[3]{343^{2}}}$

You can take the cubed root and square the numbers in either order but if you can do the root first that is often easier.

$\frac{5^{2}}{7^{2}}$

$\frac{25}{49}$

This is the answer. Alternatively, you could have squared the numbers first before taking the cubed root.

$\frac{\sqrt[3]{15625}}{\sqrt[3]{117649}}$

$\frac{25}{49}$

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Example Question #4 : Rational Exponents

Evaluate:

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}$

Possible Answers:

$-\frac{1}{4}$

Correct answer:

Explanation:

In order to evaluate fractional exponents, we can express them using the following relationship:

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

In this formula, represents the index of the radical from the denominator of the fraction and is the exponent that raises the base: . When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$

We can then rewrite and solve the expression in the following way.

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}=\frac{1}{\frac{1}{\sqrt[4]{256}}}=\frac{1}{\frac{1}{4}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{1}{\frac{1}{4}}=1\times \frac{4}{1}=4$

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Example Question #1 : Rational Exponents

Evaluate.

$144^{1/2}$

Possible Answers:

$\pm22$

$\pm12$

$\pm36$

Correct answer:

$\pm12$

Explanation:

Exponents raised to a power of <1 can be written as the root of the denominator.

So:

$144^{1/2} = \sqrt[2]{144}=\pm12$

Recall that a square root can give two answers, one positive and one negative.

$\\(-12)(-12)=144 \\(12)(12)=144$

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Example Question #1 : Rational Exponents

Evaluate

$8^{11/5}$

Possible Answers:

2.4

2.2

2.8

2.6

2.1

Correct answer:

2.2

Explanation:

$8^{11/5}$

The denominator of the exponent "N" is the same as the "N" root of that number.

$8^{11/5} = \sqrt[5]{8^{11}} = 2.2$

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Example Question #1 : Rational Exponents

Evaluate

$(7^{5/3})^{3/4}$

Possible Answers:

11.386

14.215

13.569

10.582

11.478

Correct answer:

11.386

Explanation:

This appears more complicated than it is.

$(7^{5/3})^{3/4}$ is really just X^Y where

$X=7^{5/3}$

and

y=3/4

Re-written this equation looks like

$(\sqrt[3]{7^5})^{3/4} = 1.667^{3/4}= \sqrt[4]{1.667^3} = 11.386$

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Example Question #1 : Rational Exponents

Evaluate

$17^{0.824}$

Possible Answers:

14.578

13.325

9.876

18.256

10.325

Correct answer:

10.325

Explanation:

$17^{0.824}$ can be seen as X^Y , in a scientific calculator use the X^Y button where Y=0.824 .

$17^{0.824}=10.325$

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Example Question #2 : Rational Exponents

Evaluate the given rational exponent:

$8^{\frac{2}{3}}$

Possible Answers:

Correct answer:

Explanation:

Rational exponents can be simplified by following this common rule:

$a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$

We can apply this concept to the given value in order to evaluate:

$\\8^{\frac{2}{3}}=\sqrt[3]{8^{2}}=\sqrt[3]{64}\\\sqrt[3]{64}=4$

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College Algebra : Rational Exponents

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #10 : Fractional Exponents

Example Question #1 : Rational Exponents

Example Question #11 : Fractional Exponents

Example Question #3 : Rational Exponents

Example Question #4 : Rational Exponents

Example Question #1 : Rational Exponents

Example Question #1 : Rational Exponents

Example Question #1 : Rational Exponents

Example Question #1 : Rational Exponents

Example Question #2 : Rational Exponents

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