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High School Math : Radicals

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Example Questions

Example Question #6 : Simplifying Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

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

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

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

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

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

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 #2 : Factoring Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

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

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

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

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

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

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 #21 : Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

$9-10\sqrt{3}$

$27-10\sqrt{3}$

$18-7\sqrt{3}$

$18-10\sqrt{3}$

$3-10\sqrt{3}$

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 #1 : Factoring Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

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

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

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

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

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

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 #3 : Factoring Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

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

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

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

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

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

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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Example Question #22 : Radicals

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{3}}+\sqrt{75}-2\sqrt{3}$

Possible Answers:

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

$\frac{8\sqrt{3}}{3}$

$\frac{10\sqrt{3}}{3}$

$\frac{11\sqrt{3}}{3}$

$\frac{9\sqrt{3}}{3}$

Correct answer:

$\frac{10\sqrt{3}}{3}$

Explanation:

Begin by factoring the radicals and combining like terms:

$\sqrt{\frac{1}{3}}+\sqrt{75}-2\sqrt{3}$

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

$\sqrt{\frac{1}{3}}+3\sqrt{3}$

Multiply the left side of the equation by $\frac{\sqrt{3}}{\sqrt{3}}$ and the right side by $\frac{3}{3}$ :

${\frac{1}{\sqrt{3}}}+3\sqrt{3}$

${\frac{1}{\sqrt{3}}}\cdot \frac{\sqrt{3}}{\sqrt{3}}+3\sqrt{3}\cdot\frac{3}{3}$

$\frac{\sqrt{3}}{3}+\frac{9\sqrt{3}}{3}$

$\frac{10\sqrt{3}}{3}$

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Example Question #11 : Factoring Radicals

Simplify the following radical expression:

$\sqrt[4]{64x^5y^6z^9}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by factoring the integer:

$\sqrt[4]{64x^5y^6z^9}$

$\sqrt[4]{2^6x^5y^6z^9}$

Factor the exponents:

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

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Example Question #11 : Factoring Radicals

Simplify the following radical expression:

$\sqrt{72a^5b^7c^6}$

Possible Answers:

$6a^2bc^3\sqrt{2ab}$

$6a^2b^3c\sqrt{2ab}$

$6ab^3c^3\sqrt{2ab}$

$6a^2b^3c^3\sqrt{2ab}$

$6a^2b^3c^3\sqrt{ab}$

Correct answer:

$6a^2b^3c^3\sqrt{2ab}$

Explanation:

Begin by factoring the integer:

$\sqrt{72a^5b^7c^6}$

$\sqrt{2^33^2a^5b^7c^6}$

Factor the exponents:

$6a^2b^3c^3\sqrt{2ab}$

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Example Question #21 : Radicals

Simplify the following radical expression:

$\sqrt[3]{-40n^7m^4}$

Possible Answers:

$2n^2m\sqrt[3]{5nm}$

$-n^2m\sqrt[3]{5nm}$

$-2n^2m\sqrt[3]{5nm}$

$-2n^2m\sqrt[3]{6nm}$

$n^2m\sqrt[3]{5nm}$

Correct answer:

$-2n^2m\sqrt[3]{5nm}$

Explanation:

Begin by factoring the integer:

$\sqrt[3]{-40n^7m^4}$

$\sqrt[3]{-2^35n^7m^4}$

Factor the exponents:

$-2n^2m\sqrt[3]{5nm^}$

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Example Question #11 : Simplifying Radicals

Simplify the following radical expression:

$\sqrt{n^2-10n+25}$

Possible Answers:

n-12

n-8

n-5

n-10

n-6

Correct answer:

n-5

Explanation:

Begin by factoring the expression:

$\sqrt{n^2-10n+25}$

$\sqrt{(n-5)^2}$

Now, take the square root:

n-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 #6 : Simplifying Radicals

Example Question #2 : Factoring Radicals

Example Question #21 : Radicals

Example Question #1 : Factoring Radicals

Example Question #3 : Factoring Radicals

Example Question #22 : Radicals

Example Question #11 : Factoring Radicals

Example Question #11 : Factoring Radicals

Example Question #21 : Radicals

Example Question #11 : Simplifying Radicals

All High School Math Resources

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