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GED Math : Square Roots and Radicals

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Example Questions

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Example Question #201 : Complex Operations

Factor: $3\sqrt{88}$

Possible Answers:

$9\sqrt{22}$

$12\sqrt{22}$

$5\sqrt{22}$

$\textup{The expression is already factored to its simplest form.}$

$6\sqrt{22}$

Correct answer:

$6\sqrt{22}$

Explanation:

In order to factor this, we will need to rewrite root 88 by factoring using values of perfect squares.

$3\sqrt{88} = 3\times\sqrt{4}\times \sqrt{22} = 3\times 2 \times \sqrt{22}$

The answer is: $6\sqrt{22}$

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Example Question #532 : Numbers And Operations

Rationalize: $\frac{3}{2\sqrt2}$

Possible Answers:

$\frac{\sqrt2}{4}$

$\frac{\sqrt2}{3}$

$\frac{3\sqrt{2}}{4}$

$\frac{\sqrt2}{2}$

$\frac{3\sqrt2}{2}$

Correct answer:

$\frac{3\sqrt{2}}{4}$

Explanation:

In order to rationalize the radical, we will need to multiply both the top and bottom by square root two.

$\frac{3}{2\sqrt2}\cdot\frac{\sqrt2}{\sqrt2} = \frac{3\sqrt2}{2\cdot 2}$

The answer is: $\frac{3\sqrt{2}}{4}$

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Example Question #11 : Square Roots And Radicals

Rationalize: $\frac{2}{\sqrt5}$

Possible Answers:

$\sqrt{10}$

$\frac{2\sqrt{5}}{5}$

$2\sqrt{5}$

$\frac{5\sqrt{2}}{2}$

$\frac{2\sqrt{5}}{10}$

Correct answer:

$\frac{2\sqrt{5}}{5}$

Explanation:

In order to eliminate the radical on the denominator, we will need to multiply root five on the top and bottom of the fraction.

$\frac{2}{\sqrt5}\cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

The answer is: $\frac{2\sqrt{5}}{5}$

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Example Question #11 : Square Roots And Radicals

Rationalize: $\frac{\sqrt3}{\sqrt{20}}$

Possible Answers:

$\frac{\sqrt{15}}{10}$

$\frac{\sqrt{30}}{20}$

$3\sqrt{3}$

$5\sqrt3$

$\frac{\sqrt{30}}{10}$

Correct answer:

$\frac{\sqrt{15}}{10}$

Explanation:

Simplify the denominator by factoring using perfect squares.

$\sqrt{20} = \sqrt4 \cdot \sqrt5 = 2\sqrt{5}$

Rewrite the fraction.

$\frac{\sqrt3}{\sqrt{20}}=\frac{\sqrt3}{2\sqrt{5}}$

Multiply the top and bottom by root 5.

$\frac{\sqrt3}{2\sqrt{5}} \cdot \frac{\sqrt5}{\sqrt5} = \frac{\sqrt{15}}{10}$

The answer is: $\frac{\sqrt{15}}{10}$

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Example Question #11 : Square Roots And Radicals

Rationalize: $\frac{2}{\sqrt{12}}$

Possible Answers:

$\frac{2\sqrt3}{5}$

$6\sqrt2$

$6\sqrt3$

$\frac{\sqrt3}{3}$

$\frac{2\sqrt3}{3}$

Correct answer:

$\frac{\sqrt3}{3}$

Explanation:

Factor the denominator by factors of perfect squares.

$\sqrt{12} = \sqrt4 \cdot \sqrt3 = 2\sqrt{3}$

Replace the term.

$\frac{2}{\sqrt{12}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt3}$

Multiply by root three on the top and bottom.

$\frac{1}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3} = \frac{\sqrt3}{3}$

The answer is: $\frac{\sqrt3}{3}$

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Example Question #11 : Square Roots And Radicals

Rationalize: $\frac{7\sqrt2}{\sqrt{14}}$

Possible Answers:

$\sqrt{14}$

$\frac{\sqrt7}{2}$

$\frac{4\sqrt{2}}{7}$

$\sqrt7$

$\frac{\sqrt{28}}{7}$

Correct answer:

$\sqrt7$

Explanation:

Multiply by root 14 on the top and bottom of the fraction.

$\frac{7\sqrt2}{\sqrt{14}}\cdot\frac{\sqrt{14}}{\sqrt{14}} = \frac{7\sqrt{28}}{14}$

The fraction becomes: $\frac{\sqrt{28}}{2}$

Simplify the radical with factors of perfect squares.

$\frac{\sqrt{28}}{2} = \frac{\sqrt{4}\cdot \sqrt{7}}{2} = \frac{2\cdot \sqrt{7}}{2}$

Reduce the fraction.

The answer is: $\sqrt7$

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Example Question #11 : Square Roots And Radicals

Rationalize the radical: $\frac{9}{\sqrt3}$

Possible Answers:

$\sqrt3$

$3\sqrt3$

$9\sqrt3$

$\frac{\sqrt3}{3}$

Correct answer:

$3\sqrt3$

Explanation:

Multiply the radical on the top and bottom of the given fraction.

$\frac{9}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3} = \frac{9\sqrt3}{3}$

Simplify the fraction.

The answer is: $3\sqrt3$

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Example Question #12 : Square Roots And Radicals

Rationalize: $\frac{6}{5\sqrt6}$

Possible Answers:

$\frac{3\sqrt6}{5}$

$\frac{\sqrt3}{5}$

$\frac{\sqrt6}{30}$

$\frac{2\sqrt6}{5}$

$\frac{\sqrt6}{5}$

Correct answer:

$\frac{\sqrt6}{5}$

Explanation:

Multiply the radical on the top and bottom of the fraction.

$\frac{6}{5\sqrt6} \cdot \frac{\sqrt6}{\sqrt6} = \frac{6\sqrt6}{5(6)}$

Reduce the fraction.

The answer is: $\frac{\sqrt6}{5}$

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Example Question #13 : Square Roots And Radicals

Rationalize: $\frac{5\sqrt2}{\sqrt3}$

Possible Answers:

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

$\frac{\sqrt{30}}{3}$

$\frac{5\sqrt6}{3}$

$\frac{3\sqrt6}{5}$

$15\sqrt2$

Correct answer:

$\frac{5\sqrt6}{3}$

Explanation:

Multiply the denominator on the top and bottom.

$\frac{5\sqrt2}{\sqrt3}\cdot \frac{\sqrt3}{\sqrt3} = \frac{5\sqrt6}{3}$

The answer is: $\frac{5\sqrt6}{3}$

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Example Question #13 : Square Roots And Radicals

Simplify the following expression:

$7\sqrt{3}+\sqrt{75}+\sqrt{243}$

Possible Answers:

$21\sqrt{3}$

$\sqrt{3}+5\sqrt{6}$

$14\sqrt{3}+2\sqrt{2}$

$12\sqrt{3}+12$

Correct answer:

$21\sqrt{3}$

Explanation:

Start by analyzing each given term.

$7\sqrt{3}$ cannot be reduced any further, so leave it alone.

Notice that $\sqrt{75}$ can be rewritten as $\sqrt{25(3)}=5\sqrt3$

Next, notice that $\sqrt{243}$ can be rewritten as $\sqrt{(81)(3)}=9\sqrt{3}$

Now, rewrite the original equation:

$7\sqrt{3}+\sqrt{75}+\sqrt{243}=7\sqrt{3}+5\sqrt{3}+9\sqrt{3}=21\sqrt{3}$

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GED Math : Square Roots and Radicals

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #201 : Complex Operations

Example Question #532 : Numbers And Operations

Example Question #11 : Square Roots And Radicals

Example Question #11 : Square Roots And Radicals

Example Question #11 : Square Roots And Radicals

Example Question #11 : Square Roots And Radicals

Example Question #11 : Square Roots And Radicals

Example Question #12 : Square Roots And Radicals

Example Question #13 : Square Roots And Radicals

Example Question #13 : Square Roots And Radicals

All GED Math Resources

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