Factoring Radicals

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Math › Factoring Radicals

Questions 1 - 10

1

Factor and simplify the following radical expression:

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

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

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

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

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

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

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:

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

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

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

2

Factor and simplify the following radical expression:

$(4-2\sqrt{2})(\sqrt{2}-4)$

$12\sqrt{2}-20$

$10\sqrt{2}-20$

$12\sqrt{2}-18$

$12\sqrt{2}-16$

$14\sqrt{2}-20$

Explanation

Begin by using the FOIL method (First Outer Inner Last) to expand the expression.

$(4-2\sqrt{2})(\sqrt{2}-4)$

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

Now, combine like terms:

$12\sqrt{2}-20$

3

Factor and simplify the following radical expression:

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

$4\sqrt{3}-7$

$5\sqrt{3}-7$

$6\sqrt{3}-7$

$3\sqrt{3}-7$

$2\sqrt{3}-7$

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:

$4\sqrt{3}-7$

4

Factor and simplify the following radical expression:

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

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

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

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

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

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

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 $\frac{5}{5}$

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

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

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

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

5

Simplify the following radical expression:

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

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

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

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

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

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

Explanation

Begin by factoring the integer:

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

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

Factor the exponents:

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

6

Simplify the following radical expression:

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

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

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

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

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

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

Explanation

Begin by factoring the integer:

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

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

Factor the exponents:

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

7

Factor and simplify the following radical expression:

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

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

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

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

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

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

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:

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

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

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

8

Factor and simplify the following radical expression:

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

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

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

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

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

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

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:

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

9

Simplify the following radical expression:

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

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

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

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

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

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

Explanation

Begin by factoring the integer:

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

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

Factor the exponents:

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

10

Simplify the following radical expression:

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

Explanation

Begin by factoring the expression:

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

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

Now, take the square root:

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