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Algebra II : Multiplying and Dividing Radicals

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Example Questions

Example Question #1 : Multiplying And Dividing Radicals

Multiply and express the answer in the simplest form:

$\small \sqrt3(\sqrt4-\sqrt3)$

Possible Answers:

$\small 2\sqrt3-3$

$\small 4\sqrt3-9$

$\small \sqrt7-\sqrt6$

$\small \sqrt{12}-\sqrt9$

Correct answer:

$\small 2\sqrt3-3$

Explanation:

$\small \sqrt3(\sqrt4-\sqrt3)$

$\small \sqrt{12}-\sqrt9$

$\small \sqrt{12}=\sqrt4 \times \sqrt3=2\sqrt3$

$\small \sqrt9=3$

$\small \sqrt{12}-\sqrt9=2\sqrt3-3$

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Example Question #1 : Multiplying And Dividing Radicals

$\left ( \sqrt{x}+\sqrt{6} \right )\left ( \sqrt{x}-\sqrt{6} \right )$

Possible Answers:

$x-\sqrt{12}$

$\sqrt{x}-6$

x-6

$x^{2}-36$

$x-\sqrt{6}$

Correct answer:

x-6

Explanation:

FOIL with difference of squares. The multiplying cancels the square roots on both terms.

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Example Question #4 : Radicals

Simplify.

$\sqrt{36} \times 3\sqrt{36}$

Possible Answers:

108

$4\sqrt{36}$

$3\sqrt{36}$

Correct answer:

108

Explanation:

We can solve this by simplifying the radicals first: $\sqrt{36} =6$

Plugging this into the equation gives us:

$6 \times3(6)=108$

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Example Question #3 : Multiplying And Dividing Radicals

Simplify.

$5\sqrt{2} \times4\sqrt{2}$

Possible Answers:

$20\sqrt{2}$

$\sqrt{80}$

Correct answer:

Explanation:

Note: the product of the radicals is the same as the radical of the product:

$\sqrt{2} \times \sqrt2} = \sqrt{2\times 2$ which is $\sqrt{4} =2$

Once we understand this, we can plug it into the equation:

$5\sqrt{2} \times4\sqrt{2} = 5\times4\times \sqrt{2\times 2}$

$5\times4\times2=40$

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Example Question #4 : Multiplying And Dividing Radicals

Simplify.

$\frac{\sqrt{32}}{3\sqrt{16}}$

Possible Answers:

$\frac{4}{12}$

$\sqrt{2}$

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

$\frac{4\sqrt{2}}{12}$

Correct answer:

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

Explanation:

We can simplify the radicals:

$\sqrt{32} =\sqrt{16\times 2} =4\sqrt{2}$ and $\sqrt{16} =4$

Plug in the simplifed radicals into the equation:

$\frac{4\sqrt{2}}{3\times 4} = \frac{4\sqrt{2}}{12}$

$\frac{4 \sqrt2}{12}=\frac{\sqrt2}{3}$

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Example Question #5 : Multiplying And Dividing Radicals

Simplify and rationalize the denominator if needed,

$\frac{\sqrt{16}}{4\sqrt{2}}$

Possible Answers:

$\sqrt{2}$

$\frac{\sqrt2}{2}$

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

$\frac{1}{2}$

Correct answer:

$\frac{\sqrt2}{2}$

Explanation:

We can only simplify the radical in the numerator: $\sqrt{16} =4$

Plugging in the simplifed radical into the equation we get:

$\frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}}$

Note: We simplified further because both the numerator and denominator had a "4" which canceled out.

Now we want to rationalize the denominator,

$\frac{1}{\sqrt2}\bigg(\frac{\sqrt2}{\sqrt2}\bigg)=\frac{\sqrt2}{2}$

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Example Question #1 : Multiplying And Dividing Radicals

Simplify $2x^{2}\cdot x^{5}\cdot 3x^{3}$

Possible Answers:

$5x^{10}$

$6x^{10}$

$6x^{30}$

$5x^{30}$

$6x^{13}$

Correct answer:

$6x^{10}$

Explanation:

To simplify, you must use the Law of Exponents.

First you must multiply the coefficients then add the exponents:

$2\cdot3\cdot x^{2+5+3}=6\cdot x^{10}=6x^{10}$ .

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Example Question #3 : Radicals

What is the product of $\sqrt{12}$ and $3\sqrt{6}$ ?

Possible Answers:

$18\sqrt{2}$

$9\sqrt{24}$

$3\sqrt{72}$

$6\sqrt{18}$

$12\sqrt{18}$

Correct answer:

$18\sqrt{2}$

Explanation:

First, simplify $\sqrt{12}=\sqrt{4\cdot3}$ to $2\sqrt{3}$ .

Then set up the multiplication problem:

$2\sqrt{3}\cdot 3\sqrt{6}$ .

Multiply the terms outside of the radical, then the terms under the radical:

$2\cdot 3\sqrt{3\cdot6}$ then simplfy: $6\sqrt{18}.$

The radical is still not in its simplest form and must be reduced further:

$6\sqrt{18}=6\cdot3\sqrt{2}=18\sqrt{2}$ . This is the radical in its simplest form.

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Example Question #2 : Multiplying And Dividing Radicals

Simplify

${}\frac{\sqrt{117}}{\sqrt{13}}$

Possible Answers:

$\sqrt9$

$3\sqrt{13}$

$3\sqrt9$

$\sqrt{13}$

Correct answer:

Explanation:

To divide the radicals, simply divide the numbers under the radical and leave them under the radical:

$\sqrt{\frac{117}{13}}=\sqrt9$

Then simplify this radical:

$\sqrt9=3$ .

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Example Question #1 : Multiplying And Dividing Radicals

Solve and simplify.

$\sqrt{2}\cdot \sqrt{3}$

Possible Answers:

$\sqrt{23}$

$\sqrt{6}$

$\sqrt{7}$

$\sqrt{5}$

Correct answer:

$\sqrt{6}$

Explanation:

When multiplying radicals, just take the values inside the radicand and perfom the operation.

$\sqrt{2}\cdot \sqrt{3}=\sqrt{2\cdot 3}=\sqrt{6}$

$\sqrt{6}$ can't be reduced so this is the final answer.

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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 #1 : Multiplying And Dividing Radicals

Example Question #1 : Multiplying And Dividing Radicals

Example Question #4 : Radicals

Example Question #3 : Multiplying And Dividing Radicals

Example Question #4 : Multiplying And Dividing Radicals

Example Question #5 : Multiplying And Dividing Radicals

Example Question #1 : Multiplying And Dividing Radicals

Example Question #3 : Radicals

Example Question #2 : Multiplying And Dividing Radicals

Example Question #1 : Multiplying And Dividing Radicals

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