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High School Math : Simplifying Radicals

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Example Questions

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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Example Question #31 : Radicals

Simplify the following radical expression:

$\sqrt[3]{(3x-y)^5}$

Possible Answers:

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

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

$(3x-y)\sqrt[3]{(3x-y)^3}$

$(3x-y)\sqrt[3]{(3x-y)}$

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

Correct answer:

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

Explanation:

Simplify the radical expression:

$\sqrt[3]{(3x-y)^5}$

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

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

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Example Question #31 : Radicals

Simplify the expression:

$\frac{3\sqrt[4]{32}}{2\sqrt[4]{162}}$

Possible Answers:

. $\frac{3\sqrt[4]{16}}{2\sqrt[4]{81}}$

$\frac{2}{3}$

$6\sqrt[4]{2}$

$\frac{3}{2}$

Correct answer:

Explanation:

Use the multiplication property of radicals to split the fourth roots as follows:

$\rightarrow \frac{3\sqrt[4]{16}\sqrt[4]{2}}{2\sqrt[4]{81}\sqrt[4]{2}}$

Simplify the new roots:

$\rightarrow \frac{3(2)\sqrt[4]{2}}{2(3)\sqrt[4]{2}}$

$\rightarrow \frac{6\sqrt[4]{2}}{6\sqrt[4]{2}}$

$\rightarrow 1$

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Example Question #11 : Simplifying Radicals

Find the value of $\sqrt{80} + \sqrt{20}$ .

Possible Answers:

$6\sqrt{10}$

$\sqrt5$

$4\sqrt5$

$6\sqrt5$

Correct answer:

$6\sqrt5$

Explanation:

To solve this equation, we have to factor our radicals. We do this by finding numbers that multiply to give us the number within the radical.

$\sqrt{80} = \sqrt{20}\sqrt{4}$

$\sqrt{20} = \sqrt{5}\sqrt{4}$

Add them together:

$\sqrt5\sqrt4\sqrt4 + \sqrt5\sqrt4$

4 is a perfect square, so we can find the root:

$(2)(2)\sqrt5 + 2\sqrt5$

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

Since both have the same radical, we can combine them:

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

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Example Question #12 : Simplifying Radicals

Factor and simplify the following radical expression:

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

Possible Answers:

$12\sqrt{2}-18$

$10\sqrt{2}-20$

$12\sqrt{2}-20$

$14\sqrt{2}-20$

$12\sqrt{2}-16$

Correct answer:

$12\sqrt{2}-20$

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:

$12\sqrt{2}-20$

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Example Question #13 : Simplifying Radicals

What is the value of $5\sqrt5 + 3 - 4\sqrt{2} +9 - 3\sqrt{5}$ ?

Possible Answers:

$4\sqrt{7} - 12$

$-2\sqrt{3} + 12$

$2\sqrt{5} - 4\sqrt{2} + 12$

$2\sqrt{10} - 4\sqrt{2} + 12$

$2\sqrt{7} - 4\sqrt{7} + 12$

Correct answer:

$2\sqrt{5} - 4\sqrt{2} + 12$

Explanation:

When combining terms involving radicals, we can only combine the ones that have the same radical. For this problem, that means $4\sqrt2$ has to stay on its own while we can combine $5\sqrt5$ and $-3\sqrt5$ into $2\sqrt5$ . The simple integers can be combined too, giving us our answer with three seperate terms.

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High School Math : Simplifying Radicals

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #22 : Radicals

Example Question #11 : Factoring Radicals

Example Question #11 : Factoring Radicals

Example Question #21 : Radicals

Example Question #11 : Simplifying Radicals

Example Question #31 : Radicals

Example Question #31 : Radicals

Example Question #11 : Simplifying Radicals

Example Question #12 : Simplifying Radicals

Example Question #13 : Simplifying Radicals

All High School Math Resources

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