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Algebra II : Fractional Exponents

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Example Questions

Example Question #51 : Fractional Exponents

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

Possible Answers:

$\frac{1}{4}$

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

$\frac{\sqrt{2}}{4}$

$\sqrt{2}$

$-\sqrt{2}$

Correct answer:

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

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 .

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

Remember when getting rid of radicals, we just multiply top and bottom by the radical.

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Example Question #52 : Fractional Exponents

Simplify:

$(9x)^{\frac{2}{3}}+(10x)^{\frac{3}{2}}$

Possible Answers:

$3\sqrt[3]{3x^2}+10x\sqrt{10x}$

$3\sqrt[3]{3x^2}-10x\sqrt{10x}$

$3x\sqrt[3]{3x}+10x\sqrt{10x}$

$3\sqrt[3]{9x^2}+10x\sqrt{10x}$

Correct answer:

$3\sqrt[3]{3x^2}+10x\sqrt{10x}$

Explanation:

To simplify fractional exponents, remember that the numerator of the fraction corresponds to the power the term is taken, and the denominator indicates what root we are taking of that term (i.e. 2 means square root, 3 means cube root and so on).

Doing this, we get

$\sqrt[3]{(9x)^2}+\sqrt{(10x)^3}$

Expanding the inside of the roots, we get

$\sqrt[3]{81x^2}+\sqrt{1000x^3}$

We can now factor the terms on the inside, pulling out any cubes or squares, respectively:

$\sqrt[3]{27\cdot 3\cdot x^2} +\sqrt{100\cdot 10\cdot x^2\cdot x}=3\sqrt[3]{3x^2}+10x\sqrt{10x}$

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Example Question #53 : Fractional Exponents

Write $x^{2/5}$ in radical form.

Possible Answers:

$5\sqrt{x}$

$\frac{1}{\sqrt[5]{x^{2}}}$

$2\sqrt[5]{x}$

$\sqrt[5]{x^{2}}$

$\sqrt[]{x^{5}}$

Correct answer:

$\sqrt[5]{x^{2}}$

Explanation:

Fractional exponents are an alternate way to write a radical expression. The numerator of the fractional exponent becomes the exponent of the term inside the radical and the denominator of the fractional exponent determines the index of the radical.

So $x^{2/5}$ becomes $\sqrt[5]{x^{2}}$ .

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Example Question #51 : Fractional Exponents

Simplify $7x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}$ .

Possible Answers:

$8x^{\frac{11}{12}}$

$7x^{\frac{11}{12}}$

This cannot be simplified.

$7x^{\frac{1}{6}}$

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

Correct answer:

$7x^{\frac{11}{12}}$

Explanation:

When multiplying terms with exponents, you must add the exponents.

In order to add the exponents, you need to find a common denominator, in this case the common denominator is 12.

$7x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}=7x^{\frac{8}{12}}\cdot x^{\frac{3}{12}}=7x^{\frac{11}{12}}$

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Example Question #55 : Fractional Exponents

Simplify $(216x^{6})^{-\frac{4}{3}}$ .

Possible Answers:

$\frac{1}{(x\sqrt[4]{216x^{2}})^{3}}$

$1296x^{8}$

$\frac{1}{1296x^{8}}$

$\frac{1}{1296x^{6}}$

$1296x^{6}$

Correct answer:

$\frac{1}{1296x^{8}}$

Explanation:

First, eliminate the negative exponent by placing the term in the denominator:

$(216x^{6})^{-\frac{4}{3}}=\frac{1}{(216x^{6})^{\frac{4}{3}}}$

With fractional exponents, the denominator is the index of the radical and the numerator is the power the term is raised to:

$\frac{1}{(216x^{6})^{\frac{4}{3}}}=\frac{1}{(\sqrt[3]{216x^{6}})^{4}}$

Take the cube root of the denominator and raise to the 4th power:

$\frac{1}{(6x^{2})^{4}}=\frac{1}{1296x^{8}}$

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Example Question #56 : Fractional Exponents

Simplify $(16x^{8})^{\frac{5}{2}}$ .

Possible Answers:

$1024x^{9}$

$4x^{20}$

$64x^{20}$

$1024x^{20}$

$\sqrt[5]{4096x^{16}}$

Correct answer:

$1024x^{20}$

Explanation:

With a fractional exponent, the denominator tells you the index of the radical and the numerator tells you what power to raise the term to. So this problem can be rewritten as:

$(16x^{8})^{\frac{5}{2}}=(\sqrt{16x^{8}})^{5}$

Now take the square root and raise to the 5th power:

$(\sqrt{16x^{8}})^{5}=(4x^{^{4}})^{5}=1024x^{20}$

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Example Question #57 : Fractional Exponents

Solve: $2^{-3}\times 3^{-2}$

Possible Answers:

$\frac{1}{36}$

$\frac{1}{7776}$

$\frac{1}{32768}$

$\frac{1}{144}$

$\frac{1}{72}$

Correct answer:

$\frac{1}{72}$

Explanation:

Before we can multiply the terms, convert both terms with negative exponents to fractions.

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

Rewrite the terms and simplify.

$2^{-3}\times 3^{-2} = \frac{1}{2^3}\times\frac{1}{3^2} = \frac{1}{8}\times \frac{1}{9}$

Multiply the denominators together.

The answer is: $\frac{1}{72}$

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Example Question #172 : Exponents

Simplify: $4^{\frac{3}{2}}+ 8^{\frac{2}{3}}$

Possible Answers:

$\textup{The answer is not given.}$

Correct answer:

Explanation:

Evaluate the first term by rewriting the term out as a product of exponents.

$4^{\frac{3}{2}} =(4^3)^{\frac{1}{2}} = 64^{\frac{1}{2}}$

The power of a half is similar to the square root of the number.

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

Evaluate the second term.

$8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = (\sqrt[3]{8})^2 = 2^2 =4$

Add both numbers.

8+4=12

The answer is:

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Example Question #59 : Fractional Exponents

Evaluate $8^{\frac{2}{5}}$

Possible Answers:

$\sqrt[5]{64}$

$10\sqrt{2}$

$\sqrt[5]{8}$

$\sqrt{8^5}$

$2\sqrt[5]{8}$

Correct answer:

$\sqrt[5]{64}$

Explanation:

When dealing with fractional exponents, we write as

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

which is the index of the radical, is the exponent raising base .

Therefore we have

$8^\frac{2}{5}=\sqrt[5]{8^2}=\sqrt[5]{64}$

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Example Question #53 : Fractional Exponents

Evaluate $6^{-\frac{1}{2}}$

Possible Answers:

$\sqrt6$

$-\sqrt6$

$-\frac{\sqrt6}{6}$

$\frac{\sqrt6}{6}$

$-\frac{1}{6}$

Correct answer:

$\frac{\sqrt6}{6}$

Explanation:

When dealing with fractional exponents, we write as

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

which is the index of the radical, is the exponent raising base .

We evaluate negative exponents as

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

which is the positive exponent raising base .

Therefore,

$6^{-\frac{1}{2}}=\frac{1}{\sqrt6}=\frac{1*\sqrt6}{\sqrt6*\sqrt6}=\frac{\sqrt6}{6}$

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Algebra II : Fractional Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #51 : Fractional Exponents

Example Question #52 : Fractional Exponents

Example Question #53 : Fractional Exponents

Example Question #51 : Fractional Exponents

Example Question #55 : Fractional Exponents

Example Question #56 : Fractional Exponents

Example Question #57 : Fractional Exponents

Example Question #172 : Exponents

Example Question #59 : Fractional Exponents

Example Question #53 : Fractional Exponents

All Algebra II Resources

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