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Algebra II : Fractional Exponents

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Example Question #21 : Fractional Exponents

Evaluate $16^{\frac{3}{4}}$

Possible Answers:

$\frac{\sqrt[4]{16}}{3}$

$\sqrt[4]{16}$

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.

$16^{\frac{3}{4}}=\sqrt[4]{16^3}$ We can actually get an integer value.

$\sqrt[4]{16^3}=\sqrt[4]{(2^4)^3}=\sqrt[4]{(2^3)^4}$ For the power rule of exponents, they are interchangeable because according to the definition of commutative property, multiplying two different numbers in two different positions will generate the same answer. The fourth powers will cancel leaving us with 2^3 or .

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Example Question #22 : Fractional Exponents

Evaluate $\frac{1}{125}^{-\frac{1}{3}}$

Possible Answers:

$-\frac{1}{5}$

$\frac{1}{5}$

125

Correct answer:

Explanation:

When dealing with fractional exponents, remember this form:

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.

$\frac{1}{125}^{-\frac{1}{3}}=\frac{1}{\frac{1}{125}^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{\frac{1}{125}}}=\frac{1}{\frac{1}{5}}=5$

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Example Question #23 : Fractional Exponents

Evaluate $10^{-\frac{1}{2}}$

Possible Answers:

$\frac{\sqrt{10}}{10}$

$\frac{1}{100}$

$\sqrt{10}$

$10\sqrt{10}$

Correct answer:

$\frac{\sqrt{10}}{10}$

Explanation:

When dealing with fractional exponents, remember this form:

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.

$10^{-\frac{1}{2}}=\frac{1}{10^{\frac{1}{2}}}=\frac{1}{\sqrt[2]{10}}=\frac{1}{\sqrt{10}}=\frac{\sqrt{10}}{10}$ To get rid of the radical, just multiply top and bottom by the radical.

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Example Question #24 : Fractional Exponents

$81^{\frac{3}{2}}$

Possible Answers:

121.5

729

Correct answer:

729

Explanation:

A fractional exponent means to take the number to the power of the numerator and to the root of the number in the denominator.

Take 81 to the power of 3 and to the root of 2 (square root). These can be done in either order but it is usually easier to take the root first.

$\sqrt{81^{3}}$

$9^{3}$

$9\cdot 9\cdot 9$

729

Alternatively, this is what it would look like cubing 81 first.

$\sqrt{81\cdot 81\cdot 81}$

$\sqrt{531441}$

729

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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 #25 : Fractional Exponents

Evaluate $16^{\frac{1}{4}}$ .

Possible Answers:

$\frac{1}{2}$

$\frac{1}{4}$

Correct answer:

Explanation:

When an exponent is a fraction, we convert to radical form.

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

is the exponent from the numerator of the fraction, is the index from the denominator of the fraction and is the base of the exponent.

$16^{\frac{1}{4}}=\sqrt[4]{16^1}=\sqrt[4]{4\cdot 4}=\sqrt[4]{2\cdot 2\cdot 2\cdot 2}=2$

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Example Question #21 : Fractional Exponents

Evaluate $72^{\frac{1}{2}}$ .

Possible Answers:

$2\sqrt{6}$

$7\sqrt{2}$

$6\sqrt{2}$

Correct answer:

$6\sqrt{2}$

Explanation:

When an exponent is a fraction, we convert to radical form.

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

is the exponent from the numerator of the fraction, is the index from the denominator of the fraction and is the base of the exponent.

$72^{\frac{1}{2}}=\sqrt{72}=\sqrt{36*2}=\sqrt{6\cdot 6\cdot 2}=6\sqrt{2}$

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Example Question #27 : Fractional Exponents

Evaluate $-27^{\frac{1}{3}}$ .

Possible Answers:

$-\frac{1}{3}$

Does not exist

Correct answer:

Explanation:

Although we see a negative number there, keep in mind the fractional exponent.

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

is the exponent from the numerator of the fraction, is the index from the denominator of the fraction and is the base of the exponent.

We know it's a cubic root and any number cubed can generate negative answers.

In general, any odd exponent can generate negative answers.

$-27^{\frac{1}{3}}=\sqrt[3]{-27}=-3$

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Example Question #28 : Fractional Exponents

Evaluate $\frac{1}{5}^{\frac{1}{2}}$

Possible Answers:

$-\frac{1}{5}$

$\frac{\sqrt{5}}{5}$

$\frac{\sqrt{5}}{2}$

$\sqrt{5}$

$\frac{1}{5}$

Correct answer:

$\frac{\sqrt{5}}{5}$

Explanation:

When an exponent is a fraction, we convert to radical form.

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

is the exponent from the numerator of the fraction, is the index from the denominator of the fraction and is the base of the exponent.

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

To eliminate the radical in the denominator, remember to multiply top and bottom by itself.

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Example Question #29 : Fractional Exponents

Evaluate $\frac{3}{7}^{\frac{2}{3}}$ .

Possible Answers:

$\sqrt{\frac{2}{3}}$

$\sqrt{\frac{3}{7}}$

$\sqrt[3]{\frac{9}{49}}$

$\frac{9}{49}$

$\sqrt[3]{\frac{3}{7}}$

Correct answer:

$\sqrt[3]{\frac{9}{49}}$

Explanation:

When an exponent is a fraction, we convert to radical form.

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

is the exponent from the numerator of the fraction, is the index from the denominator of the fraction and is the base of the exponent.

$\frac{3}{7}^{\frac{2}{3}}=\sqrt[3]{(\frac{3}{7})^2}=\sqrt[3]{\frac{9}{49}}$

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Algebra II : Fractional Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Fractional Exponents

Example Question #22 : Fractional Exponents

Example Question #23 : Fractional Exponents

Example Question #24 : Fractional Exponents

Example Question #3 : Rational Exponents

Example Question #25 : Fractional Exponents

Example Question #21 : Fractional Exponents

Example Question #27 : Fractional Exponents

Example Question #28 : Fractional Exponents

Example Question #29 : Fractional Exponents

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