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Algebra II : Radicals as Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #71 : Radicals

Rewrite the following radical as an exponent:

$\sqrt[2]{xyz^{2}}$

Possible Answers:

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

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

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

xyz

Correct answer:

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

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:

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

From this point simplify the exponents accordingly:

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

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Example Question #71 : Radicals

Rewrite the following radical as an exponent:

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

Possible Answers:

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

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

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

$x^{3}yz^{2}$

Correct answer:

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

Explanation:

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

From this point simplify the exponents accordingly:

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

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Example Question #3971 : Algebra Ii

Rewrite the following radical as an exponent:

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

Possible Answers:

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

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

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

Correct answer:

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

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 both terms by multiplying it by the exponents of each term as shown below:

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

From this point simplify the exponents accordingly:

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

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Example Question #21 : Radicals As Exponents

Rewrite the following radical as an exponent:

$\sqrt[3]{y^{3}z^{7}}$

Possible Answers:

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

$yz^{7}$

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

Correct answer:

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

Explanation:

$\sqrt[3]{y^{3}z^{7}}=(y^{3}z^{7})^{\frac{1}{3}}=y^{\frac{3}{3}}z^{\frac{7}{3}}$

From this point simplify the exponents accordingly:

$y^{\frac{3}{3}}z^{\frac{7}{3}}=yz^{\frac{7}{3}}$

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Example Question #21 : Radicals As Exponents

Rewrite the following radical as an exponent:

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

Possible Answers:

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

$x^{8}$

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

$x^{2}$

Correct answer:

$x^{2}$

Explanation:

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

From this point simplify the exponents accordingly:

$x^{\frac{4}{2}}=x^{2}$

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Example Question #73 : Radicals

Evaluate: $64^{\frac{1}{6}}+8^{\frac{2}{3}}$

Possible Answers:

Correct answer:

Explanation:

In order to add the two terms, we must first find the values of each term in the expression.

Rewrite the fractional exponents as an expression of a square root.

$64^{\frac{1}{6}}=\sqrt[6]{64}=2$

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

Add the two values.

2+4=6

The answer is:

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Example Question #22 : Radicals As Exponents

$Rewrite\ as\ an\ exponential\ function:\ \sqrt{x}$

Possible Answers:

$x^{\sqrt{x}}$

$x^{2}$

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

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

Correct answer:

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

Explanation:

Fractional exponents have the power as the numerator and the root as the denominator.

$\sqrt{x}\ has\ an\ understood\ power\ of\ 1\ and\ an\ understood\ root\ of\ 2.$

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

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Example Question #23 : Radicals As Exponents

$Rewrite\ with\ a\ radical: x^\frac{5}{3}$

Possible Answers:

$\sqrt[5]{x^3}$

$\sqrt{x^\frac{5}{3}}$

$\sqrt[3]{x^5}$

x^1^5

Correct answer:

$\sqrt[3]{x^5}$

Explanation:

Fractional exponents have the power as the numerator and the root as the denominator.

In this case, the power is 5, and the root is 3.

$When\ taking\ the\ 3rd\ root\ of\ x\ to\ the\ 5th\ power\ it\ is\ written\ as:$

$x^\frac{5}{3}=\sqrt[3]{x^5}$

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Example Question #78 : Radicals

Solve: $144^\frac{3}{2}-64^\frac{1}{6}$

Possible Answers:

142

1730

1726

345

3371

Correct answer:

1726

Explanation:

The numbers with the fractional exponents can be rewritten as radicals.

$144^\frac{3}{2}-64^\frac{1}{6}=( \sqrt{144})^3 - \sqrt[6]{64}$

Simplify both radicals. The sixth root of 64 is two.

12^3 -2 = 1728-2 = 1726

The answer is: 1726

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Example Question #24 : Radicals As Exponents

Solve: $-64^{\frac{1}{4}}$

Possible Answers:

$-4i\sqrt2$

$4i\sqrt2$

$2i\sqrt2$

$-2\sqrt2$

$\textup{There is no solution.}$

Correct answer:

$-2\sqrt2$

Explanation:

This can be rewritten as the fourth root.

$-64^{\frac{1}{4}} = -\sqrt[4]{64}$

Rewrite the inner quantity with factors of numbers to the fourth root.

1^4 =1, 2^4 = 16

$-\sqrt[4]{64} = -\sqrt[4]{16\times 4} = -\sqrt[4]{16} \cdot \sqrt[4]{4}$

Simplify both terms.

$-\sqrt[4]{16} =-2$

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

The answer is: $-2\sqrt2$

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Algebra II : Radicals as Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #71 : Radicals

Example Question #71 : Radicals

Example Question #3971 : Algebra Ii

Example Question #21 : Radicals As Exponents

Example Question #21 : Radicals As Exponents

Example Question #73 : Radicals

Example Question #22 : Radicals As Exponents

Example Question #23 : Radicals As Exponents

Example Question #78 : Radicals

Example Question #24 : Radicals As Exponents

All Algebra II Resources

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