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SAT Math : How to multiply square roots

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Example Questions

Example Question #1 : How To Multiply Square Roots

Simplify: $\sqrt{7}*\sqrt{14}$ $\displaystyle \sqrt{7}*\sqrt{14}$

Possible Answers:

$\sqrt{98}$ $\displaystyle \sqrt{98}$

$7\sqrt{2}$ $\displaystyle 7\sqrt{2}$

$7\sqrt{3}$ $\displaystyle 7\sqrt{3}$

$2\sqrt{7}$ $\displaystyle 2\sqrt{7}$

$\sqrt{21}$ $\displaystyle \sqrt{21}$

Correct answer:

$7\sqrt{2}$ $\displaystyle 7\sqrt{2}$

Explanation:

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{7}*\sqrt{14}=\sqrt{7*14}=\sqrt{98}=\sqrt{49}*\sqrt{2}=7\sqrt{2}$ $\displaystyle \sqrt{7}*\sqrt{14}=\sqrt{7*14}=\sqrt{98}=\sqrt{49}*\sqrt{2}=7\sqrt{2}$

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Example Question #1 : How To Multiply Square Roots

Simplify: $\sqrt{10}*\sqrt{15}$ $\displaystyle \sqrt{10}*\sqrt{15}$

Possible Answers:

$75\sqrt{2}$ $\displaystyle 75\sqrt{2}$

$10\sqrt{2}$ $\displaystyle 10\sqrt{2}$

$5\sqrt{6}$ $\displaystyle 5\sqrt{6}$

$\sqrt{150}$ $\displaystyle \sqrt{150}$

$6\sqrt{5}$ $\displaystyle 6\sqrt{5}$

Correct answer:

$5\sqrt{6}$ $\displaystyle 5\sqrt{6}$

Explanation:

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{10}*\sqrt{15}=\sqrt{10*15}=\sqrt{150}=\sqrt{25}*\sqrt{6}=5\sqrt{6}$ $\displaystyle \sqrt{10}*\sqrt{15}=\sqrt{10*15}=\sqrt{150}=\sqrt{25}*\sqrt{6}=5\sqrt{6}$

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Example Question #1 : Square Roots And Operations

Simplify: $\sqrt{24}*3\sqrt{8}$ $\displaystyle \sqrt{24}*3\sqrt{8}$

Possible Answers:

$8\sqrt{14}$ $\displaystyle 8\sqrt{14}$

$16\sqrt{6}$ $\displaystyle 16\sqrt{6}$

$24\sqrt{3}$ $\displaystyle 24\sqrt{3}$

$12\sqrt{3}$ $\displaystyle 12\sqrt{3}$

$24\sqrt{6}$ $\displaystyle 24\sqrt{6}$

Correct answer:

$24\sqrt{3}$ $\displaystyle 24\sqrt{3}$

Explanation:

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{24}*3\sqrt{8}=3\sqrt{24*8}=3\sqrt{192}=3\sqrt{64}*\sqrt{3}=24\sqrt{3}$ $\displaystyle \sqrt{24}*3\sqrt{8}=3\sqrt{24*8}=3\sqrt{192}=3\sqrt{64}*\sqrt{3}=24\sqrt{3}$

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Example Question #2 : Square Roots And Operations

Simplify:

$(\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{6})$ $\displaystyle (\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{6})$

Possible Answers:

$\sqrt{6}+\sqrt{18}+\sqrt{10}+\sqrt{30}+8$ $\displaystyle \sqrt{6}+\sqrt{18}+\sqrt{10}+\sqrt{30}+8$

$\sqrt{18}+\sqrt{10}$ $\displaystyle \sqrt{18}+\sqrt{10}$

$4\sqrt{6}+5\sqrt{2}+\sqrt{3}$ $\displaystyle 4\sqrt{6}+5\sqrt{2}+\sqrt{3}$

$\sqrt{6}+\sqrt{30}$ $\displaystyle \sqrt{6}+\sqrt{30}$

$\sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$ $\displaystyle \sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$

Correct answer:

$\sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$ $\displaystyle \sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$

Explanation:

To simplify the problem, just distribute the radical to each term in the parentheses.

$(\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{6})=$ $\displaystyle (\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{6})=$

$\sqrt{6}+\sqrt{18}+\sqrt{10}+\sqrt{30}=$ $\displaystyle \sqrt{6}+\sqrt{18}+\sqrt{10}+\sqrt{30}=$

$\sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$ $\displaystyle \sqrt{6}+3\sqrt{2}+\sqrt{10}+\sqrt{30}$

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Example Question #1 : How To Multiply Square Roots

Evaluate and simplify:

$\sqrt{10}*\sqrt{12}$ $\displaystyle \sqrt{10}*\sqrt{12}$

Possible Answers:

$4\sqrt{5}$ $\displaystyle 4\sqrt{5}$

$20\sqrt{6}$ $\displaystyle 20\sqrt{6}$

$\sqrt{22}$ $\displaystyle \sqrt{22}$

$2\sqrt{30}$ $\displaystyle 2\sqrt{30}$

$30\sqrt{2}$ $\displaystyle 30\sqrt{2}$

Correct answer:

$2\sqrt{30}$ $\displaystyle 2\sqrt{30}$

Explanation:

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

$\\ \sqrt{10}*\sqrt{12}\\=\sqrt{10*12}\\=\sqrt{120}\\=\sqrt{4}*\sqrt{30}\\=2\sqrt{30}$ $\displaystyle \\ \sqrt{10}*\sqrt{12}\\=\sqrt{10*12}\\=\sqrt{120}\\=\sqrt{4}*\sqrt{30}\\=2\sqrt{30}$

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Example Question #2 : Square Roots And Operations

Simplify and evaluate:

$\sqrt{12}*\sqrt{48}$ $\displaystyle \sqrt{12}*\sqrt{48}$

Possible Answers:

$\displaystyle 24$

$8\sqrt{3}$ $\displaystyle 8\sqrt{3}$

$\displaystyle 18$

$16\sqrt{2}$ $\displaystyle 16\sqrt{2}$

$24\sqrt{3}$ $\displaystyle 24\sqrt{3}$

Correct answer:

$\displaystyle 24$

Explanation:

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

In this case, let's simplify each individual radical and multiply them.

$\\\sqrt{12}*\sqrt{48}\\=\sqrt{4}*\sqrt{3}*\sqrt{16}*\sqrt{3}\\=2\sqrt{3}*4\sqrt{3}\\=8*3=24$ $\displaystyle \\\sqrt{12}*\sqrt{48}\\=\sqrt{4}*\sqrt{3}*\sqrt{16}*\sqrt{3}\\=2\sqrt{3}*4\sqrt{3}\\=8*3=24$

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Example Question #1 : How To Multiply Square Roots

Simplify: $\sqrt{x+5}*\sqrt{x-5}$ $\displaystyle \sqrt{x+5}*\sqrt{x-5}$

Possible Answers:

$\sqrt{x^2-25}$ $\displaystyle \sqrt{x^2-25}$

x-5 $\displaystyle x-5$

x+5 $\displaystyle x+5$

x^2-25 $\displaystyle x^2-25$

$\sqrt{x^2-10}$ $\displaystyle \sqrt{x^2-10}$

Correct answer:

$\sqrt{x^2-25}$ $\displaystyle \sqrt{x^2-25}$

Explanation:

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

$\\\sqrt{x+5}*\sqrt{x-5}\\=\sqrt{(x+5)(x-5)}\\=\sqrt{x^2-25}$ $\displaystyle \\\sqrt{x+5}*\sqrt{x-5}\\=\sqrt{(x+5)(x-5)}\\=\sqrt{x^2-25}$

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Example Question #1 : How To Multiply Square Roots

Simplify:

$\sqrt{6}*2\sqrt{3}$ $\displaystyle \sqrt{6}*2\sqrt{3}$

Possible Answers:

$6\sqrt{2}$ $\displaystyle 6\sqrt{2}$

$6\sqrt{3}$ $\displaystyle 6\sqrt{3}$

$12\sqrt{2}$ $\displaystyle 12\sqrt{2}$

$4\sqrt{6}$ $\displaystyle 4\sqrt{6}$

$2\sqrt{6}$ $\displaystyle 2\sqrt{6}$

Correct answer:

$6\sqrt{2}$ $\displaystyle 6\sqrt{2}$

Explanation:

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied. We can simplify them if possible.

$\\\sqrt{6}*2\sqrt{3}\\=2\sqrt{6*3}\\=2\sqrt{18}\\=2\sqrt{9}*\sqrt{2}\\=2*3\sqrt{2}\\=6\sqrt{2}$ $\displaystyle \\\sqrt{6}*2\sqrt{3}\\=2\sqrt{6*3}\\=2\sqrt{18}\\=2\sqrt{9}*\sqrt{2}\\=2*3\sqrt{2}\\=6\sqrt{2}$

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Example Question #3 : How To Multiply Square Roots

Simplify:

$4\sqrt{7}*\sqrt{21}$ $\displaystyle 4\sqrt{7}*\sqrt{21}$

Possible Answers:

$3\sqrt{7}$ $\displaystyle 3\sqrt{7}$

$11\sqrt{3}$ $\displaystyle 11\sqrt{3}$

$12\sqrt{21}$ $\displaystyle 12\sqrt{21}$

$28\sqrt{3}$ $\displaystyle 28\sqrt{3}$

$7\sqrt{7}$ $\displaystyle 7\sqrt{7}$

Correct answer:

$28\sqrt{3}$ $\displaystyle 28\sqrt{3}$

Explanation:

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied.

We can simplify them if possible.

$\\4\sqrt{7}*\sqrt{21}\\ =4\sqrt{7*21}\\=4\sqrt{7*7*3}\\=4*7\sqrt{3}\\=28\sqrt{3}$ $\displaystyle \\4\sqrt{7}*\sqrt{21}\\ =4\sqrt{7*21}\\=4\sqrt{7*7*3}\\=4*7\sqrt{3}\\=28\sqrt{3}$

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Example Question #1 : How To Multiply Square Roots

Simplify:

$2\sqrt{10}*5\sqrt{20}$ $\displaystyle 2\sqrt{10}*5\sqrt{20}$

Possible Answers:

200 $\displaystyle 200$

$100\sqrt{2}$ $\displaystyle 100\sqrt{2}$

$20\sqrt{5}$ $\displaystyle 20\sqrt{5}$

$50\sqrt{2}$ $\displaystyle 50\sqrt{2}$

$40\sqrt{10}$ $\displaystyle 40\sqrt{10}$

Correct answer:

$100\sqrt{2}$ $\displaystyle 100\sqrt{2}$

Explanation:

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied.

We can simplify them if possible.

$\\2\sqrt{10}*5\sqrt{20}\\=10\sqrt{10*20}\\=10\sqrt{10*10*2}\\=10*10\sqrt{2}\\=100\sqrt{2}$ $\displaystyle \\2\sqrt{10}*5\sqrt{20}\\=10\sqrt{10*20}\\=10\sqrt{10*10*2}\\=10*10\sqrt{2}\\=100\sqrt{2}$

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SAT Math : How to multiply square roots

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Multiply Square Roots

Example Question #1 : How To Multiply Square Roots

Example Question #1 : Square Roots And Operations

Example Question #2 : Square Roots And Operations

Example Question #1 : How To Multiply Square Roots

Example Question #2 : Square Roots And Operations

Example Question #1 : How To Multiply Square Roots

Example Question #1 : How To Multiply Square Roots

Example Question #3 : How To Multiply Square Roots

Example Question #1 : How To Multiply Square Roots

All SAT Math Resources

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