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High School Math : Radicals

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Example Question #10 : Expressing Radicals As Exponents

Simplify: $(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

Possible Answers:

$\frac{a^{\frac{1}{2}}b^{\frac{1}{2}}}{2b}$

$\frac{ab^{\frac{1}{2}}}{4b}$

$\frac{a^{\frac{1}{2}}b}{4b}$

$\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

Correct answer:

$\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

Explanation:

Multiply the numerator and denominator to the exponent:

$(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

$=\frac{2^{-2}a^{\frac{-1}{2}}}{a^{-1}b^{\frac{1}{2}}}$

Simplify the expression by combining like terms:

$=\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

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Example Question #11 : Expressing Radicals As Exponents

Express the following exponent in radical form:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by converting each exponent to have a denominator of :

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

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

Now, rearrange into radical form:

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

Finally, simplify:

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

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Example Question #12 : Expressing Radicals As Exponents

Express the following exponent in radical form:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by converting each exponent to have a denominator of :

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

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

Now, put this in radical form:

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

Finally, simplify:

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

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Example Question #13 : Expressing Radicals As Exponents

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

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

Possible Answers:

$3n\sqrt{3n}$

$2n\sqrt{3n}$

$2n\sqrt{2n}$

$n\sqrt{2n}$

$n\sqrt{3n}$

Correct answer:

$n\sqrt{2n}$

Explanation:

Begin by converting the radical into exponent form:

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

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

Simplify the exponent and multiply:

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

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

Convert into radical form:

$\sqrt{2n^3}$

Simplify:

$n\sqrt{2n}$

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Example Question #11 : Radicals

Express the following exponent in radical form:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

Now, put this in radical form and simplify:

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

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

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

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Example Question #1 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt[4]{81x^6y^5z^{10}}$

Possible Answers:

$3x^2yz^2\sqrt[4]{x^2yz^2}$

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

$3xyz\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2y^2z^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

Correct answer:

$3xyz^2\sqrt[4]{x^2yz^2}$

Explanation:

Begin by factoring the integer:

$\sqrt[4]{81x^6y^5z^{10}}$

$\sqrt[4]{3^4x^6y^5z^{10}}$

$3\sqrt[4]{x^6y^5z^{10}}$

Now, simplify the exponents:

$3\sqrt[4]{x^4x^2y^4yz^{4}z^4z^2}$

$3xyzz\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

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Example Question #2 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt[3]{(3n-2)^3}$

Possible Answers:

n-2

(3n-2)^2

3n+2

(3n-2)^3

3n-2

Correct answer:

3n-2

Explanation:

Begin by converting the radical into exponent form:

$\sqrt[3]{(3n-2)^3}$

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

Now, multiply the exponents:

3n-2

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Example Question #3 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

Possible Answers:

$3abc\sqrt[4]{6ab^3c^2}$

$3abc\sqrt[4]{6a^2b^3c^2}$

$3abc^2\sqrt[4]{6ab^3c^2}$

$3ab^2c\sqrt[4]{6ab^3c^2}$

$3a^2bc\sqrt[4]{6ab^3c^2}$

Correct answer:

$3abc\sqrt[4]{6ab^3c^2}$

Explanation:

Begin by converting the radical into exponent form:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

$(18a^3bc^5)^{\frac{1}{4}}\cdot ({27a^2b^6c})^{\frac{1}{4}}$

Now, combine the bases:

$(486a^5b^7c^6)^{\frac{1}{4}}$

Simplify the integer:

$(2\cdot 3\cdot 3^4a^5b^7c^6)^{\frac{1}{4}}$

Now, simplify the exponents:

$(2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4)^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\sqrt[4]{2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4}$

$3abc\sqrt[4]{2\cdot 3ab^3c^2}$

$3abc\sqrt[4]{6ab^3c^2}$

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Example Question #4 : Factoring Radicals

Factor and simplify the following radical expression:

$\frac{3-2\sqrt{3}}{2\sqrt{3}+3}$

Possible Answers:

$4\sqrt{3}-7$

$2\sqrt{3}-7$

$5\sqrt{3}-7$

$6\sqrt{3}-7$

$3\sqrt{3}-7$

Correct answer:

$4\sqrt{3}-7$

Explanation:

Begin by multiplying the numerator and denominator by the complement of the denominator:

$\frac{3-2\sqrt{3}}{2\sqrt{3}+3} \cdot \frac{2\sqrt{3}-3}{2\sqrt{3}-3}$

Use the FOIL method to multiply the radicals. F (first) O (outer) I (inner) L (last)

$\frac{6\sqrt{3}-9-12+6\sqrt{3}}{12-6\sqrt{3}+6\sqrt{3}-9}$

Now, combine like terms:

$\frac{12\sqrt{3}-21}{3}$

Simplify:

$4\sqrt{3}-7$

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Example Question #5 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{5}}-2\sqrt{20}+3\sqrt{5}$

Possible Answers:

$\frac{-4\sqrt{5}}{3}$

$\frac{-4\sqrt{5}}{5}$

$\frac{-4\sqrt{3}}{3}$

$\frac{-4\sqrt{3}}{5}$

$\frac{4\sqrt{5}}{3}$

Correct answer:

$\frac{-4\sqrt{5}}{5}$

Explanation:

Begin by factoring the radicals:

$\sqrt{\frac{1}{5}}-2\sqrt{20}+3\sqrt{5}$

$\sqrt{\frac{1}{5}}-4\sqrt{5}+3\sqrt{5}$

Combine like terms:

$\sqrt{\frac{1}{5}}-\sqrt{5}$

Multiply the left side by $\frac{\sqrt{5}}{\sqrt{5}}$ and the right side by $\frac{5}{5}$

$\frac{1}{\sqrt{5}}-\sqrt{5}$

$\frac{1}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}-\sqrt{5}\cdot\frac{5}{5}$

$\frac{\sqrt{5}}{5}-\frac{5\sqrt{5}}{5}$

$\frac{-4\sqrt{5}}{5}$

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High School Math : Radicals

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #10 : Expressing Radicals As Exponents

Example Question #11 : Expressing Radicals As Exponents

Example Question #12 : Expressing Radicals As Exponents

Example Question #13 : Expressing Radicals As Exponents

Example Question #11 : Radicals

Example Question #1 : Factoring Radicals

Example Question #2 : Factoring Radicals

Example Question #3 : Factoring Radicals

Example Question #4 : Factoring Radicals

Example Question #5 : Factoring Radicals

All High School Math Resources

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