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High School Math : Expressing Radicals as Exponents

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Example Questions

Example Question #51 : Mathematical Relationships And Basic Graphs

Express the following exponent in radical form:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by converting each exponent to have a denominator of $\displaystyle 6$:

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

$=x^{\frac{2}{6}}y^{\frac{3}{6}}z^{\frac{9}{6}}$ $\displaystyle =x^{\frac{2}{6}}y^{\frac{3}{6}}z^{\frac{9}{6}}$

Now, rearrange into radical form:

$=\sqrt[6]{x^2y^3z^9}$ $\displaystyle =\sqrt[6]{x^2y^3z^9}$

Finally, simplify:

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

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Example Question #11 : Expressing Radicals As Exponents

Express the following exponent in radical form:

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$ $\displaystyle r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

Possible Answers:

$r^4\sqrt[6]{s^2y^3}$ $\displaystyle r^4\sqrt[6]{s^2y^3}$

$r^6\sqrt[6]{s^2y^3}$ $\displaystyle r^6\sqrt[6]{s^2y^3}$

$r\sqrt[6]{s^2y^3}$ $\displaystyle r\sqrt[6]{s^2y^3}$

$r^2\sqrt[6]{s^2y^3}$ $\displaystyle r^2\sqrt[6]{s^2y^3}$

$r^3\sqrt[6]{s^2y^3}$ $\displaystyle r^3\sqrt[6]{s^2y^3}$

Correct answer:

$r^2\sqrt[6]{s^2y^3}$ $\displaystyle r^2\sqrt[6]{s^2y^3}$

Explanation:

Begin by converting each exponent to have a denominator of $\displaystyle 6$:

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$ $\displaystyle r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

$=r^{\frac{12}{6}}s^{\frac{2}{6}}y^{\frac{3}{6}}$ $\displaystyle =r^{\frac{12}{6}}s^{\frac{2}{6}}y^{\frac{3}{6}}$

Now, put this in radical form:

$=\sqrt[6]{r^{12}s^2y^3}$ $\displaystyle =\sqrt[6]{r^{12}s^2y^3}$

Finally, simplify:

$=r^2\sqrt[6]{s^2y^3}$ $\displaystyle =r^2\sqrt[6]{s^2y^3}$

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Example Question #11 : Expressing Radicals As Exponents

Simplify the following radical expression using exponents. Express the final answer in radical form.

$\sqrt[6]{8n^9}$ $\displaystyle \sqrt[6]{8n^9}$

Possible Answers:

$2n\sqrt{2n}$ $\displaystyle 2n\sqrt{2n}$

$3n\sqrt{3n}$ $\displaystyle 3n\sqrt{3n}$

$2n\sqrt{3n}$ $\displaystyle 2n\sqrt{3n}$

$n\sqrt{3n}$ $\displaystyle n\sqrt{3n}$

$n\sqrt{2n}$ $\displaystyle n\sqrt{2n}$

Correct answer:

$n\sqrt{2n}$ $\displaystyle n\sqrt{2n}$

Explanation:

Begin by converting the radical into exponent form:

$\sqrt[6]{8n^9}$ $\displaystyle \sqrt[6]{8n^9}$

$(8^{\frac{1}{6}}n^{\frac{9}{6}})$ $\displaystyle (8^{\frac{1}{6}}n^{\frac{9}{6}})$

Simplify the exponent and multiply:

$(2^3^{\frac{1}{6}}n^{\frac{9}{6}})$ $\displaystyle (2^3^{\frac{1}{6}}n^{\frac{9}{6}})$

$(2^{\frac{1}{2}}n^{\frac{3}{2}})$ $\displaystyle (2^{\frac{1}{2}}n^{\frac{3}{2}})$

Convert into radical form:

$\sqrt{2n^3}$ $\displaystyle \sqrt{2n^3}$

Simplify:

$n\sqrt{2n}$ $\displaystyle n\sqrt{2n}$

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Example Question #51 : Mathematical Relationships And Basic Graphs

Express the following exponent in radical form:

$(3x)^{\frac{1}{2}}x^{\frac{1}{4}}$ $\displaystyle (3x)^{\frac{1}{2}}x^{\frac{1}{4}}$

Possible Answers:

$\sqrt[4]{9x^2}$ $\displaystyle \sqrt[4]{9x^2}$

$\sqrt[4]{9x^3}$ $\displaystyle \sqrt[4]{9x^3}$

$\sqrt[2]{7x^3}$ $\displaystyle \sqrt[2]{7x^3}$

$\sqrt[4]{7x^3}$ $\displaystyle \sqrt[4]{7x^3}$

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

Correct answer:

$\sqrt[4]{9x^3}$ $\displaystyle \sqrt[4]{9x^3}$

Explanation:

Begin by changing the fractional exponents so that they both have a common denominator of $\displaystyle 4$:

$(3x)^{\frac{1}{2}}x^{\frac{1}{4}}$ $\displaystyle (3x)^{\frac{1}{2}}x^{\frac{1}{4}}$

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

Now, put this in radical form and simplify:

$=\sqrt[4]{(3x)^2x}$ $\displaystyle =\sqrt[4]{(3x)^2x}$

$=\sqrt[4]{9x^2x}$ $\displaystyle =\sqrt[4]{9x^2x}$

$=\sqrt[4]{9x^3}$ $\displaystyle =\sqrt[4]{9x^3}$

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High School Math : Expressing Radicals as Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #51 : Mathematical Relationships And Basic Graphs

Example Question #11 : Expressing Radicals As Exponents

Example Question #11 : Expressing Radicals As Exponents

Example Question #51 : Mathematical Relationships And Basic Graphs

All High School Math Resources

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