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Algebra II : Fractional Exponents

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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 #41 : Fractional Exponents

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

Possible Answers:

$\sqrt[8]{3}$

$\frac{3}{8}$

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

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

$\sqrt{3^8}$

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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Example Question #42 : Fractional Exponents

Simplify: $9^{\frac{1}{9}}$

Possible Answers:

$-\sqrt[9]{9}$

$\frac{1}{\sqrt[9]{9}}$

$\sqrt[9]{9}$

Correct answer:

$\sqrt[9]{9}$

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 .

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

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Example Question #43 : Fractional Exponents

Simplify: $16^{-\frac{2}{3}}$

Possible Answers:

$-\sqrt[3]{256}$

$\sqrt[3]{256}$

$-\frac{1}{\sqrt[3]{256}}$

$\frac{1}{\sqrt[3]{256}}$

Correct answer:

$\frac{1}{\sqrt[3]{256}}$

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 . When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$16^{-\frac{2}{3}}=\frac{1}{\sqrt[3]{16^2}}=\frac{1}{\sqrt[3]{256}}$

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Example Question #44 : Fractional Exponents

Simplify: $20^{-\frac{2}{5}}$

Possible Answers:

$\frac{1}{\sqrt[5]{400}}$

$-\frac{1}{8}$

$-\sqrt[5]{400}$

$\sqrt[5]{400}$

Correct answer:

$\frac{1}{\sqrt[5]{400}}$

Explanation:

$20^{-\frac{2}{5}}=\frac{1}{\sqrt[5]{20^2}}=\frac{1}{\sqrt[5]{400}}$

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Example Question #45 : Fractional Exponents

Simplify: $\frac{1}{4}^{\frac{1}{2}}$

Possible Answers:

$\frac{1}{8}$

$\frac{1}{12}$

$\frac{1}{16}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

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 .

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

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Example Question #46 : Fractional Exponents

Simplify:

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

where a=3x^4y^8

Possible Answers:

$x^2y^4\sqrt[4]{3}$

$xy\sqrt[4]{3}$

$81x^{16}y^{32}$

$xy^2\sqrt[4]{3}$

Correct answer:

$xy^2\sqrt[4]{3}$

Explanation:

When dealing with fractional exponents, remember that the numerator of the fraction represents the power to which we are taking the term that has the exponent, and the denominator represents the degree of the root we are taking of that term.

For our expression, the numerator is 1, which means we raise a to the first power. The denominator is 4, which means we are taking the fourth root of the term:

$\sqrt[4]{(3x^4y^8)^1}$

We can only move the cubes out of the radical, and when we do so, we get

$xy^2\sqrt[4]{3}$

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Example Question #47 : Fractional Exponents

Evaluate $17^\frac{1}{2}$

Possible Answers:

$\sqrt{17}$

$\sqrt{8.5}$

8.5

Correct answer:

$\sqrt{17}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$17^\frac{1}{2}=\sqrt{17}$

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Example Question #48 : Fractional Exponents

Evaluate $18^\frac{2}{3}$

Possible Answers:

$\sqrt[3]{324}$

$\sqrt[3]{54}$

$8\sqrt{2}$

$\sqrt{5832}$

Correct answer:

$\sqrt[3]{324}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$18^\frac{2}{3}=\sqrt[3]{18^2}=\sqrt[3]{324}$

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Example Question #49 : Fractional Exponents

Evaluate $24^\frac{2}{5}$

Possible Answers:

$\sqrt[5]{676}$

$2\sqrt[5]{18}$

$24\sqrt{2}$

9.6

$2\sqrt[5]{2}$

Correct answer:

$2\sqrt[5]{18}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$24^\frac{2}{5}=\sqrt[5]{24^2}=\sqrt[5]{576}=2\sqrt[5]{18}$

We were able to simplify it by factoring out perfect fifth root.

In this case, it was 32 =2^5 .

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Algebra II : Fractional Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #4 : Rational Exponents

Example Question #41 : Fractional Exponents

Example Question #42 : Fractional Exponents

Example Question #43 : Fractional Exponents

Example Question #44 : Fractional Exponents

Example Question #45 : Fractional Exponents

Example Question #46 : Fractional Exponents

Example Question #47 : Fractional Exponents

Example Question #48 : Fractional Exponents

Example Question #49 : Fractional Exponents

All Algebra II Resources

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