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High School Math : Factoring Radicals

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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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Example Question #6 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

Possible Answers:

$2abc\sqrt[4]{72ab^2c^3}$

$2abc^2\sqrt[4]{72ab^2c^3}$

$2abc^2\sqrt[4]{72a^2b^2c^3}$

$2ab^2c^2\sqrt[4]{72ab^2c^3}$

$2a^2bc^2\sqrt[4]{72ab^2c^3}$

Correct answer:

$2abc^2\sqrt[4]{72ab^2c^3}$

Explanation:

Begin by converting the radicals into exponent form:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

$(24a^2b^3c^5)^{\frac{1}{4}}\cdot(48a^3b^3c^6)^{\frac{1}{4}}$

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

Now, combine the bases:

$(3^2\cdot 2^7a^5b^6c^{11})^{\frac{1}{4}}$

$(3^2\cdot 2^7aa^4b^2b^4c^3c^{8})^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\sqrt[4]{3^2\cdot 2^7aa^4b^2b^4c^3c^{8}}$

$2abc^2\sqrt[4]{3^22^3ab^2c^3}$

$2abc^2\sqrt[4]{72ab^2c^3}$

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Example Question #7 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

Possible Answers:

$(1+m)\sqrt{3mn}$

$(1+2m)\sqrt{2mn}$

$(1+3m)\sqrt{3mn}$

$(1+2m)\sqrt{3mn}$

$(1+m)\sqrt{mn}$

Correct answer:

$(1+3m)\sqrt{3mn}$

Explanation:

Begin by simplifying the right side of the rational expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

$\sqrt{3mn}+3m\sqrt{3mn}$

$1\sqrt{3mn}+3m\sqrt{3mn}$

Now, combine like terms:

$(1+3m)\sqrt{3mn}$

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Example Question #8 : Factoring Radicals

Factor and simplify the following radical expression:

$(4-\sqrt{3})(6-\sqrt{3})$

Possible Answers:

$9-10\sqrt{3}$

$18-7\sqrt{3}$

$27-10\sqrt{3}$

$18-10\sqrt{3}$

$3-10\sqrt{3}$

Correct answer:

$27-10\sqrt{3}$

Explanation:

Begin by using the FOIL method to multiply the radical expression. F (first) O (outer) I (inner) L (last)

$(4-\sqrt{3})(6-\sqrt{3})$

$24-4\sqrt{3}-6\sqrt{3}+3$

Now, combine like terms:

$27-10\sqrt{3}$

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Example Question #9 : Factoring Radicals

Factor and simplify the following radical expression:

$\sqrt[3]{\frac{2}{5n}}$

Possible Answers:

$\frac{\sqrt[3]{30n^2}}{3n}$

$\frac{\sqrt[3]{50n^2}}{3n}$

$\frac{\sqrt[3]{30n^2}}{5n}$

$\frac{\sqrt[3]{40n^2}}{5n}$

$\frac{\sqrt[3]{50n^2}}{5n}$

Correct answer:

$\frac{\sqrt[3]{50n^2}}{5n}$

Explanation:

Begin by multiplying the numerator and denominator by $\frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$ :

$\sqrt[3]{\frac{2}{5n}}$

$\frac{\sqrt[3]{2}}{\sqrt[3]{5n}}\cdot$ $\frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$

$\frac{\sqrt[3]{50n^2}}{5n}$

The expression cannot be further simplified.

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Example Question #10 : Factoring Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

$\frac{2-\sqrt{2}}{4+2\sqrt{2}} \cdot \frac{4-2\sqrt{2}}{4-2\sqrt{2}}$

$\frac{8-4\sqrt{2}-4\sqrt{2}+4}{16-8}$

Combine like terms and simplify:

$\frac{12-8\sqrt{2}}{8}$

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

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High School Math : Factoring Radicals

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

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

Example Question #6 : Factoring Radicals

Example Question #7 : Factoring Radicals

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Example Question #10 : Factoring Radicals

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