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Algebra II : Radicals

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Example Questions

Example Question #3 : Radicals As Exponents

Simplify:

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

Possible Answers:

$xz^2\sqrt[3]{100x^2}$

$xz\sqrt[3]{100x}$

$10x^3z^4\sqrt{10x}$

$xz^2\sqrt[3]{100x}$

Correct answer:

$xz^2\sqrt[3]{100x}$

Explanation:

To simplify the expression, we must remember that a fraction as a power denotes a radical: the numerator is the power to which the term is taken inside the radical, and the denominator denotes the degree of the root (i.e. 2 means square root, 3 means cube root, etc.)

Rewriting our expression, we get

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

which expanded becomes

$\sqrt[3]{100x^4z^6}=\sqrt[3]{100\cdot x^3\cdot x\cdot z^6}$

Now, move the cubes outside of the cube root, after taking their cube root, leaving behind the terms that aren't cubes:

$xz^2\sqrt[3]{100x}$

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Example Question #61 : Radicals

Simplify: $-\frac{1}{16^{\frac{1}{4}}}+81^{\frac{1}{4}}$

Possible Answers:

$\frac{1}{2}$

$\frac{4}{5}$

$1\frac{1}{4}$

$5\frac{3}{4}$

$2\frac{1}{2}$

Correct answer:

$2\frac{1}{2}$

Explanation:

The fractional exponent represents the fourth root of the given number.

Evaluate $16^{\frac{1}{4}}$ .

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

Evaluate $81^{\frac{1}{4}}$ .

$81^{\frac{1}{4}}=\sqrt[4]{81} = 3$

Rewrite the fractions.

$-\frac{1}{16^{\frac{1}{4}}}+81^{\frac{1}{4}} = -\frac{1}{2}+3 = 2\frac{1}{2}$

The answer is: $2\frac{1}{2}$

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Example Question #12 : Radicals As Exponents

Rewrite the following expression as a radical: $(10)^{\frac{2}{3}}$

Possible Answers:

$(\sqrt{10})^3$

$\sqrt[3]{10}$

$\sqrt{30}$

$10\sqrt{10}$

$\sqrt[3]{100}$

Correct answer:

$\sqrt[3]{100}$

Explanation:

The denominator of the exponent represents the root and the numerator represents the power of what the radical is raised to.

We can also separate the powers using the product rule of exponents.

Rewrite the expression as follows:

$(10)^{\frac{2}{3}} =(10^2)^{\frac{1}{3}}= 100^{\frac{1}{3}} = \sqrt[3]{100}$

Note that the powers in the product can also be interchanged.

$(10^{\frac{1}{3}})^2= (\sqrt[3]{10})^2$

Either answer provided is correct.

The answer is: $\sqrt[3]{100}$

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Example Question #61 : Radicals

Rewrite the radical as an exponent:

$\sqrt[5]{x^{4}}$

Possible Answers:

$x^{\frac{1}{5}}$

$x^{\frac{4}{5}}$

$x^{\frac{5}{4}}$

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

Correct answer:

$x^{\frac{4}{5}}$

Explanation:

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent as shown below:

$\sqrt[5]{x^{4}}=x^{\frac{4}{5}}$

In this case, we are done because there are no further simplification steps.

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Example Question #14 : Radicals As Exponents

Express in simplified radical form

$(2016)^{1/2} (1728)^{1/3}$

Possible Answers:

$144\sqrt{14}$

$72\sqrt{26}$

None of these

$12\sqrt{14}$

$163\sqrt{11}$

Correct answer:

$144\sqrt{14}$

Explanation:

$(2016)^{1/2} (1728)^{1/3}$

Converting to radicals

$\sqrt{2016} \sqrt[3]{1728}$

Factoring:

$\sqrt{9*16*14} \sqrt[3]{27*8*8}$

Simplifying:

$12\sqrt{14}*12$

$144\sqrt{14}$

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Example Question #62 : Radicals

Rewrite the following radical as an exponent:

$\sqrt[3]{y^{9}}$

Possible Answers:

$y^{9}$

$y^{3}$

$y^{6}$

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

Correct answer:

$y^{3}$

Explanation:

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to the term by multiplying it by the exponent of the term as shown below:

$\sqrt[3]{y^{9}}=(y^{9})^{\frac{1}{3}}=y^{\frac{9}{3}}$

From this point simplify the exponent accordingly:

$y^{\frac{9}{3}}=y^{3}$

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Example Question #16 : Radicals As Exponents

Rewrite the following radical as an exponent:

$\sqrt[3]{x^{6}y^{2}}$

Possible Answers:

$x^{\frac{2}{3}}y^{\frac{2}{3}}$

$x^{2}y^{2}$

$x^{\frac{1}{3}}y^{\frac{1}{3}}$

$x^{2}y^{\frac{2}{3}}$

Correct answer:

$x^{2}y^{\frac{2}{3}}$

Explanation:

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to both terms by multiplying it by the exponents of each term as shown below:

$\sqrt[3]{x^{6}y^{2}}=(x^{6}y^{2})^{\frac{1}{3}}=x^{\frac{6}{3}}y^{\frac{2}{3}}$

From this point simplify the exponents accordingly:

$x^{\frac{6}{3}}y^{\frac{2}{3}}=x^{2}y^{\frac{2}{3}}$

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Example Question #13 : Radicals As Exponents

Rewrite the following radical as an exponent:

$\sqrt[3]{y^{6}z}$

Possible Answers:

$y^{2}z^{3}$

$y^{2}z^{\frac{3}{2}}$

$y^{2}z^{\frac{1}{3}}$

$y^{\frac{2}{3}}z^{\frac{1}{3}}$

Correct answer:

$y^{2}z^{\frac{1}{3}}$

Explanation:

$\sqrt[3]{y^{6}z}=(y^{6}z)^{\frac{1}{3}}=y^{\frac{6}{3}}z^{\frac{1}{3}}$

From this point simplify the exponents accordingly:

$y^{\frac{6}{3}}z^{\frac{1}{3}}=y^{2}z^{\frac{1}{3}}$

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Example Question #11 : Radicals As Exponents

Rewrite the following radical as an exponent:

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

Possible Answers:

$x^{\frac{1}{3}}y^{\frac{3}{2}}z^{2}$

$x^{\frac{1}{2}}y^{3}z^{2}$

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

$x^{\frac{1}{2}}y^{\frac{3}{2}}z^{2}$

Correct answer:

$x^{\frac{1}{2}}y^{\frac{3}{2}}z^{2}$

Explanation:

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to all terms by multiplying it by the exponents of each term as shown below:

$\sqrt[2]{xy^{3}z^{4}}=(xy^{3}z^{4})^{\frac{1}{2}}=x^{\frac{1}{2}}y^{\frac{3}{2}}z^{\frac{4}{2}}$

From this point simplify the exponents accordingly:

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

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Example Question #19 : Radicals As Exponents

Rewrite the following radical as an exponent:

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

Possible Answers:

$x^{2}y^{2}$

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

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

Correct answer:

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

Explanation:

$\sqrt[4]{x^{4}y^{2}}=(x^{4}y^{2})^{\frac{1}{4}}=x^{\frac{4}{4}}y^{\frac{2}{4}}$

From this point simplify the exponents accordingly:

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

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Algebra II : Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3 : Radicals As Exponents

Example Question #61 : Radicals

Example Question #12 : Radicals As Exponents

Example Question #61 : Radicals

Example Question #14 : Radicals As Exponents

Example Question #62 : Radicals

Example Question #16 : Radicals As Exponents

Example Question #13 : Radicals As Exponents

Example Question #11 : Radicals As Exponents

Example Question #19 : Radicals As Exponents

All Algebra II Resources

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