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Algebra II : Multiplying and Dividing Radicals

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Example Question #61 : Multiplying And Dividing Radicals

Multiply the radicals: $\sqrt{5}\times -\sqrt{10} \times \sqrt2$

Possible Answers:

$-10\sqrt2$

$-\frac{3\sqrt5}{2}$

$-2\sqrt{5}$

10i

Correct answer:

Explanation:

Multiply the numbers in the radicals to combine as one radical.

$\sqrt{5}\times -\sqrt{10} \times \sqrt2= -\sqrt{100}$

This value can be simplified as a perfect square.

The answer is:

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Example Question #62 : Multiplying And Dividing Radicals

Multiply the radicals: $\sqrt{3}\times \sqrt{6}\times \sqrt{12}$

Possible Answers:

$2\sqrt6$

$6\sqrt6$

$4\sqrt3$

$6\sqrt3$

$12\sqrt3$

Correct answer:

$6\sqrt6$

Explanation:

In order to multiply these radicals, we are allowed to multiply all three integers to one radical, but the final term will need to be simplified.

Instead, we can pull out common factors in order to simplify the terms.

$\sqrt{6}=\sqrt{2} \cdot \sqrt3$

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

Rewrite the expression.

$\sqrt{3}\times \sqrt{6}\times \sqrt{12}=\sqrt{3}\times(\sqrt{2} \cdot \sqrt3)\times 2\sqrt{3}$

A radical multiplied by itself will give just the integer.

$\sqrt{3}\times(\sqrt{2} \cdot \sqrt3)\times 2\sqrt{3} = 3\times \sqrt2 \times 2\sqrt3 = 6\sqrt6$

The answer is: $6\sqrt6$

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Example Question #161 : Simplifying Radicals

Divide the radicals: $\frac{2\sqrt{6}}{\sqrt{60}}$

Possible Answers:

$\frac{4\sqrt3}{3}$

$\frac{2\sqrt{10}}{3}$

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

$\frac{\sqrt{10}}{12}$

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

Correct answer:

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

Explanation:

Rationalize the denominator by multiplying the square root of 60 on the numerator and denominator.

$\frac{2\sqrt{6}}{\sqrt{60}}\cdot \frac{\sqrt{60}}{\sqrt{60}}$

Simplify the top and bottom of the fractions.

$\frac{2\sqrt{360}}{60}$

The radical can be simplified by using common factors of perfect squares.

$\sqrt{360} = \sqrt{36}\cdot \sqrt{10} = 6\sqrt{10}$

Rewrite the term.

$\frac{2\sqrt{360}}{60} =\frac{2 \cdot 6\sqrt{10}}{60} = \frac{\sqrt{10}}{5}$

The answer is: $\frac{\sqrt{10}}{5}$

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Example Question #64 : Multiplying And Dividing Radicals

Evaluate: $\frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{14}}{\sqrt{18}}$

Possible Answers:

$\frac{1}{3}$

$\frac{\sqrt3}{3}$

$\sqrt6$

$\sqrt3$

Correct answer:

$\frac{\sqrt3}{3}$

Explanation:

Since all terms are in radicals, we can simplify the terms by using common factors.

$\frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{14}}{\sqrt{18}} = \frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{2}\cdot \sqrt{7}}{\sqrt{3}\cdot \sqrt{6}} =\frac{\sqrt{2}}{\sqrt{6}}=\frac{1}{\sqrt3}$

Rationalize the denominator.

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

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

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Example Question #65 : Multiplying And Dividing Radicals

Multiply: $\sqrt{50}\cdot \sqrt{18}\cdot \sqrt8$

Possible Answers:

$15\sqrt{2}$

$30\sqrt2$

$30\sqrt3$

$60\sqrt{2}$

$45\sqrt{3}$

Correct answer:

$60\sqrt{2}$

Explanation:

We can simplify the radicals before expanding by multiplication.

$\sqrt{50}\cdot \sqrt{18}\cdot \sqrt8 = [\sqrt{25}\cdot \sqrt2]\cdot [\sqrt{9}\cdot \sqrt{2}]\cdot [\sqrt{4}\cdot \sqrt2]$

Simplify the radicals.

$5\sqrt2 \cdot 3\sqrt2 \cdot 2\sqrt2$

Multiply the integers together. When a square root of a number is multiplied by itself, the radical will be eliminated, giving only the integer.

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

The answer is: $60\sqrt{2}$

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Example Question #61 : Multiplying And Dividing Radicals

$\frac{5+3\sqrt{2}}{-3-\sqrt{5}}=?$

Possible Answers:

$\frac{15-5\sqrt{5}-9\sqrt{2}+3\sqrt{10}}{4}$

$\frac{-15+5\sqrt{5}-9\sqrt{2}+3\sqrt{10}}{4}$

$\frac{-15-42\sqrt{10}}{4}$

$\frac{-15-5\sqrt{5}+9\sqrt{2}-3\sqrt{10}}{4}$

$\frac{-15+8\sqrt{5}-6\sqrt{2}}{4}$

Correct answer:

$\frac{-15+5\sqrt{5}-9\sqrt{2}+3\sqrt{10}}{4}$

Explanation:

To solve this expression, multiply the numerator and the denominator by the complex conjugate of the denominator. Since the denominator is $-3-\sqrt{5}$ , the complex conjugate of this is $-3+\sqrt{5}$ . Therefore:

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

$=\frac{(5+3\sqrt{2})(-3+\sqrt{5})}{(-3-\sqrt{5})(-3+\sqrt{5})}$

$=\frac{-15+5\sqrt{5}-9\sqrt{2}+3\sqrt{10}}{9-5}$

$=\frac{-15+5\sqrt{5}-9\sqrt{2}+3\sqrt{10}}{4}$

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Algebra II : Multiplying and Dividing Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #61 : Multiplying And Dividing Radicals

Example Question #62 : Multiplying And Dividing Radicals

Example Question #161 : Simplifying Radicals

Example Question #64 : Multiplying And Dividing Radicals

Example Question #65 : Multiplying And Dividing Radicals

Example Question #61 : Multiplying And Dividing Radicals

All Algebra II Resources

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