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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #31 : Fractional Exponents

Simplify:

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

Possible Answers:

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{16}$

$\frac{1}{24}$

$\frac{1}{8}$

Correct answer:

$\frac{1}{2}$

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.

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

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Example Question #3283 : Algebra Ii

Evaluate:

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

Possible Answers:

4.5

$\frac{1}{9}$

$\pm3$

$\frac{1}{81}$

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 #3285 : Algebra Ii

Evaluate:

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

Possible Answers:

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

$\frac{11}{12}$

$\frac{3}{4}$

$\frac{2401}{4096}$

$\frac{98}{125}$

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 #151 : Exponents

Evaluate:

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

Possible Answers:

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{2}$

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 #151 : 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 #152 : 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 #3287 : Algebra Ii

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 #153 : Exponents

Evaluate:

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

Possible Answers:

$-\frac{1}{9}$

$\frac{1}{9}$

$-\frac{1}{3}$

$\frac{1}{3}$

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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Example Question #5 : 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:

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

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

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 #3291 : Algebra Ii

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

Possible Answers:

$\frac{1}{\sqrt[8]{3}}$

$\sqrt[8]{3}$

$\frac{3}{8}$

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

$\sqrt{3^8}$

Correct answer:

$\sqrt[8]{3}$

Explanation:

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base .

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

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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #31 : Fractional Exponents

Example Question #3283 : Algebra Ii

Example Question #3285 : Algebra Ii

Example Question #151 : Exponents

Example Question #151 : Exponents

Example Question #152 : Exponents

Example Question #3287 : Algebra Ii

Example Question #153 : Exponents

Example Question #5 : Rational Exponents

Example Question #3291 : Algebra Ii

All Algebra II Resources

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