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GED Math : Complex Operations

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Example Questions

Example Question #3 : Square Roots And Radicals

Simplify:

$\sqrt{42}$

Do not use a calculator.

Possible Answers:

$3 \sqrt{7}$

The expression is in simplest form.

$7 \sqrt{2}$

$2 \sqrt{7}$

Correct answer:

The expression is in simplest form.

Explanation:

The prime factorization of 42 is

$42 = 2 \times 3 \times 7$ .

Since 42 is the product of distinct primes, it has no perfect square factors, and, therefore, its square root cannot be simplified further. It is already in simplifed form.

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Example Question #6 : Square Roots And Radicals

Factor: $2\sqrt{120}$

Possible Answers:

$24\sqrt5$

$24\sqrt{10}$

$4\sqrt{30}$

$12\sqrt{10}$

$2\sqrt{15}$

Correct answer:

$4\sqrt{30}$

Explanation:

In order to factor the radical, we will need to rewrite 120 as multiples of perfect squares.

$2\sqrt{120} = 2\sqrt{30} \cdot \sqrt{4}$

Reduce the known term

$2\sqrt{30} \cdot \sqrt{4} =2\sqrt{30} \cdot 2 = 4\sqrt{30}$

The value of $\sqrt{30}$ cannot be factored any further.

The answer is: $4\sqrt{30}$

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Example Question #1 : Square Roots And Radicals

Factor: $6\sqrt{40}$

Possible Answers:

$24\sqrt{10}$

$2\sqrt{30}$

$40\sqrt{3}$

$12\sqrt{10}$

$\textup{The expression is already factored.}$

Correct answer:

$12\sqrt{10}$

Explanation:

Rewrite the root 40 in factors of perfect squares.

$6\sqrt{40} = 6 \cdot \sqrt{4}\cdot \sqrt{10} = 6 \cdot 2\cdot \sqrt{10}$

The answer is: $12\sqrt{10}$

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Example Question #201 : Complex Operations

Factor: $3\sqrt{88}$

Possible Answers:

$12\sqrt{22}$

$5\sqrt{22}$

$6\sqrt{22}$

$9\sqrt{22}$

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

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 #202 : Complex Operations

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

Possible Answers:

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

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

$\frac{\sqrt2}{4}$

$\frac{\sqrt2}{3}$

$\frac{\sqrt2}{2}$

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 #201 : Complex Operations

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

Possible Answers:

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

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

$\sqrt{10}$

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

$2\sqrt{5}$

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:

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

$5\sqrt3$

$3\sqrt{3}$

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

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

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 #201 : Complex Operations

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

Possible Answers:

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

$6\sqrt2$

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

$6\sqrt3$

$\frac{\sqrt3}{3}$

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 #201 : Complex Operations

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

Possible Answers:

$\sqrt{14}$

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

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

$\sqrt7$

$\frac{\sqrt7}{2}$

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 #202 : Complex Operations

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

Possible Answers:

$3\sqrt3$

$\sqrt3$

$9\sqrt3$

$\frac{\sqrt3}{3}$

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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GED Math : Complex Operations

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #3 : Square Roots And Radicals

Example Question #6 : Square Roots And Radicals

Example Question #1 : Square Roots And Radicals

Example Question #201 : Complex Operations

Example Question #202 : Complex Operations

Example Question #201 : Complex Operations

Example Question #11 : Square Roots And Radicals

Example Question #201 : Complex Operations

Example Question #201 : Complex Operations

Example Question #202 : Complex Operations

All GED Math Resources

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