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Algebra II : Fractional Exponents

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Example Questions

Example Question #1 : Understanding Fractional Exponents

Simplify the expression:

$\small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$

Possible Answers:

1024

256

Correct answer:

256

Explanation:

Remember that fraction exponents are the same as radicals.

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

$256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

$16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4$

$\small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$

Either method, we then need to multiply to two terms.

$\small (4)(64)=256$

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Example Question #1 : Understanding Fractional Exponents

Convert the exponent to radical notation.

$x^{\frac{3}{7}}$

Possible Answers:

$\small \small \sqrt[3]{x^7}$

$\small \frac{1}{x^4}$

$\small \frac{x^3}{x^7}$

$\small \small \sqrt[7]{x^3}$

Correct answer:

$\small \small \sqrt[7]{x^3}$

Explanation:

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

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

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$

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Example Question #1 : Simplifying Expressions

Simplify:

$[x^{1/2}]^{7/3}$

Possible Answers:

$x^{5/2}$

$x^{4/3}$

$x^{7/6}$

$x^{8/6}$

$x^{17/6}$

Correct answer:

$x^{7/6}$

Explanation:

$[x^{m}]^{n}=x^{m\times n}$

$x^{1/2\times 7/3}=x^{7/6}$

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Example Question #4 : Fractional Exponents

Write the product of $\small a^\frac{3}{4}*a^\frac{3}{8}*a^\frac{5}{2}$ in radical form

Possible Answers:

$\small \sqrt[a]{8^2^9}$

$\small \sqrt[a]{29^8}$

$\small \sqrt[2]{a^5}$

$\small \sqrt[8]{a^2^9}$

$\small \sqrt[29]{a^8}$

Correct answer:

$\small \sqrt[8]{a^2^9}$

Explanation:

This problem relies on the key knowledge that $\small x^\frac{a}{b}=\sqrt[b]{x^a}$ and that the multiplying terms with exponents requires adding the exponents. Therefore, we can rewrite the expression thusly:

$\small a^\frac{3}{4}*a^\frac{3}{8}*a^\frac{5}{2}=a^{\frac{3}{4}+\frac{3}{8}+\frac{5}{2}}=a^{\frac{29}{8}}=\sqrt[8]{a^{29}}$

Therefore, $\small \sqrt[8]{a^{29}}$ is our final answer.

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Example Question #2 : Fractional Exponents

Evaluate the following expression:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}$

Possible Answers:

$\frac{3}{25}$

$\frac{9}{5}$

$\frac{3}{5}$

$\frac{3}{\sqrt{25}}$

$\sqrt{\frac{27}{125}}$

Correct answer:

$\frac{3}{5}$

Explanation:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}=\frac{3}{5}$

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}= \sqrt[3]{\frac{27}{125}}=\frac{\sqrt[3]{27}}{\sqrt[3]{125}}=\frac{3}{5}$

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Example Question #2 : Fractional Exponents

Simplify:

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}$

Possible Answers:

$\frac{x^2}{y^2}$

x^2y^2

$\frac{1}{x^2y^2}$

$\frac{y^2}{x^2}$

Correct answer:

$\frac{x^2}{y^2}$

Explanation:

Keep in mind that when you are dividing exponents with the same base, you will want to subtract the exponent found in the denominator from the exponent found in the numerator.

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

$\frac{1}{2}-(-\frac{3}{2})=2$

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

2-4=-2

Since the exponent is negative, you will want to put the in the denominator in order to make it positive.

So then,

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}=\frac{x^2}{y^2}$

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Example Question #2 : Fractional Exponents

Find the value of $9^{\frac{3}{2}}$ .

Possible Answers:

729

Correct answer:

Explanation:

When you have a number or value with a fractional exponent,

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

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

So then,

$9^{\frac{3}{2}}=\sqrt[2]{9^3}=\sqrt{729}=27$

$9^{\frac{3}{2}}=(\sqrt9)^3=3^3=27$

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Example Question #3 : Fractional Exponents

Find the value of $64^{\frac{4}{3}}$

Possible Answers:

4096

512

256

1024

Correct answer:

256

Explanation:

When you have a number or value with a fractional exponent,

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

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

So then,

$64^{\frac{4}{3}}=\sqrt[3]{64^4}=\sqrt[3]{2^{24}}=2^8=256$

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Example Question #3 : Fractional Exponents

Simplify:

$(27t^3)^\frac{5}{3}$

Possible Answers:

243t^5

$243t^{\frac{8}{3}}$

81t^5

27t^5

Correct answer:

243t^5

Explanation:

When exponents are raised to another exponent, you will need to multiply the exponents together.

When you have a number or value with a fractional exponent,

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

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

So,

$27^{\frac{5}{3}}=\sqrt[3]{27^5}=\sqrt[3]{3^{15}}=3^5=243$

$(t^3)^{\frac{5}{3}}=t^{3\times\frac{5}{3}}=t^{\frac{15}{3}}=t^5$

$(27t^3)^\frac{5}{3}=243t^5$

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Example Question #23 : Review And Other Topics

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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Algebra II : Fractional Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Understanding Fractional Exponents

Example Question #1 : Understanding Fractional Exponents

Example Question #1 : Simplifying Expressions

Example Question #4 : Fractional Exponents

Example Question #2 : Fractional Exponents

Example Question #2 : Fractional Exponents

Example Question #2 : Fractional Exponents

Example Question #3 : Fractional Exponents

Example Question #3 : Fractional Exponents

Example Question #23 : Review And Other Topics

All Algebra II Resources

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