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Algebra II : Fractional Exponents

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Example Question #31 : Fractional Exponents

Evaluate $256^{-\frac{1}{2}}$ .

Possible Answers:

$-\frac{1}{16}$

$\frac{1}{16}$

128

Correct answer:

$\frac{1}{16}$

Explanation:

When an exponent is negative, we rewrite in this form in terms of fraction:

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

is the base while is the exponent and it's going to be a postive number when written in a fraction.

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

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.

$256^{-\frac{1}{2}}=(\frac{1}{256})^{\frac{1}{2}}=\sqrt{\frac{1}{256}}=\frac{1}{16}$

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Example Question #32 : Fractional Exponents

Simplify:

$48^{\frac{1}{2}}$

Possible Answers:

$2\sqrt{3}$

$2\sqrt{6}$

$4\sqrt{3}$

Correct answer:

$4\sqrt{3}$

Explanation:

When we have fractional exponents, we convert like this:

$x^{\frac{a}b{}}=\sqrt[b]{x^a}$ . is the index of the radical which is the denominator of the fractional exponent, is the power that will raise the base which is the numerator of the fractional exponent and is the base.

$48^{\frac{1}{2}}=\sqrt{48}=\sqrt{16*3}=4\sqrt{3}$

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Example Question #33 : Fractional Exponents

Simplify:

$\frac{1}{8}^{\frac{1}{3}}$

Possible Answers:

$\frac{1}{24}$

$\frac{1}{16}$

$\frac{1}{8}$

$\frac{1}{4}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

When we have fractional exponents, we convert like this:

$\frac{1}{8}^{\frac{1}{3}}=\sqrt[3]{\frac{1}{8}}=\frac{1}{2}$

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Example Question #34 : Fractional Exponents

Evaluate:

$9^{\frac{1}{2}}$

Possible Answers:

$\frac{1}{81}$

$\frac{1}{9}$

4.5

$\pm3$

Correct answer:

$\pm3$

Explanation:

When exponents are in fraction, there are turned into this form:

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$ where represents the index of the radical which is the denominator of the fractional exponent, is the power raising base . is the numerator of the fractional exponent.

$9^{\frac{1}{2}}=\sqrt{9}=\pm3$

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Example Question #35 : Fractional Exponents

Evaluate:

$\frac{49}{64}^{\frac{1}{2}}$

Possible Answers:

$\frac{2401}{4096}$

$\pm\frac{7}{8}$

$\frac{98}{125}$

$\frac{11}{12}$

$\frac{3}{4}$

Correct answer:

$\pm\frac{7}{8}$

Explanation:

When exponents are in fraction, there are turned into this form:

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

where represents the index of the radical which is the denominator of the fractional exponent, is the power raising base . is the numerator of the fractional exponent.

$\frac{49}{64}^{\frac{1}{2}}=\sqrt{\frac{49}{64}}=\pm\frac{7}{8}$

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Example Question #36 : Fractional Exponents

Evaluate:

$16^{\frac{1}{2}}$

Possible Answers:

$\frac{1}{8}$

$\frac{1}{2}$

$\frac{1}{4}$

Correct answer:

Explanation:

When exponents are fractional, we express them using the following format:

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

In this format, is the index of the radical found in the denominator of the fractional exponent and is the exponent found in the numerator of the fractional exponent by raising the base of .

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

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Example Question #37 : Fractional Exponents

Evaluate:

$216^{\frac{2}{3}}$

Possible Answers:

Correct answer:

Explanation:

When exponents are fractional, we express them using the following format:

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

In this format, is the index of the radical found in the denominator of the fractional exponent and is the exponent found in the numerator of the fractional exponent by raising the base of .

$216^{\frac{2}{3}}=\sqrt[3]{216^2}=\sqrt[3]{(6^3)^2}=6^2=36$

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Example Question #38 : Fractional Exponents

Evaluate:

$512^{\frac{1}{3}}$

Possible Answers:

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: .

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

$512^{\frac{1}{3}}=\sqrt[3]{512^1}=8$

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Example Question #39 : Fractional Exponents

Evaluate:

$256^{\frac{1}{4}}$

Possible Answers:

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: .

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

$256^{\frac{1}{4}}=\sqrt[4]{256}=4$

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Example Question #40 : Fractional Exponents

Evaluate:

$729^{-\frac{1}{3}}$

Possible Answers:

$-\frac{1}{9}$

$-\frac{1}{3}$

$\frac{1}{9}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{9}$

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.

$729^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{729}}=\frac{1}{9}$

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Algebra II : Fractional Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #31 : Fractional Exponents

Example Question #32 : Fractional Exponents

Example Question #33 : Fractional Exponents

Example Question #34 : Fractional Exponents

Example Question #35 : Fractional Exponents

Example Question #36 : Fractional Exponents

Example Question #37 : Fractional Exponents

Example Question #38 : Fractional Exponents

Example Question #39 : Fractional Exponents

Example Question #40 : Fractional Exponents

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