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SAT Math : Arithmetic

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Example Questions

Example Question #4 : How To Multiply Square Roots

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #2 : How To Multiply Square Roots

Evaluate and simplify:

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

Possible Answers:

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

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

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

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

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

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 #1 : How To Multiply Square Roots

Simplify and evaluate:

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

Possible Answers:

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

$\displaystyle 18$

$\displaystyle 24$

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

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

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 #6 : How To Multiply Square Roots

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

Possible Answers:

x+5 $\displaystyle x+5$

x-5 $\displaystyle x-5$

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

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 #7 : How To Multiply Square Roots

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #2 : How To Multiply Square Roots

Simplify:

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

Possible Answers:

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

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

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

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

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

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:

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

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

200 $\displaystyle 200$

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

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

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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Example Question #81 : Arithmetic

Simplify:

$\sqrt{3}(\sqrt{3}+4)$ $\displaystyle \sqrt{3}(\sqrt{3}+4)$

Possible Answers:

$\displaystyle 12$

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

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

$3+4\sqrt{3}$ $\displaystyle 3+4\sqrt{3}$

$9+4\sqrt{3}$ $\displaystyle 9+4\sqrt{3}$

Correct answer:

$3+4\sqrt{3}$ $\displaystyle 3+4\sqrt{3}$

Explanation:

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

$\\\sqrt{3}(\sqrt{3}+4)\\=\sqrt{9}+4\sqrt{3}\\=3+4\sqrt{3}$ $\displaystyle \\\sqrt{3}(\sqrt{3}+4)\\=\sqrt{9}+4\sqrt{3}\\=3+4\sqrt{3}$

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Example Question #82 : Arithmetic

Simplify:

$\sqrt{3}(\sqrt{7}+\sqrt{12})$ $\displaystyle \sqrt{3}(\sqrt{7}+\sqrt{12})$

Possible Answers:

$12+4\sqrt{3}$ $\displaystyle 12+4\sqrt{3}$

$\sqrt{21}+3\sqrt{3}$ $\displaystyle \sqrt{21}+3\sqrt{3}$

$21+3\sqrt{3}$ $\displaystyle 21+3\sqrt{3}$

$\sqrt{21}+2\sqrt{2}$ $\displaystyle \sqrt{21}+2\sqrt{2}$

$\sqrt{21}+6$ $\displaystyle \sqrt{21}+6$

Correct answer:

$\sqrt{21}+6$ $\displaystyle \sqrt{21}+6$

Explanation:

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

$\\\sqrt{3}(\sqrt{7}+\sqrt{12})\\=\sqrt{21}+\sqrt{36}\\=\sqrt{21}+6$ $\displaystyle \\\sqrt{3}(\sqrt{7}+\sqrt{12})\\=\sqrt{21}+\sqrt{36}\\=\sqrt{21}+6$

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Example Question #83 : Arithmetic

Simplify:

$-\sqrt{6}(\sqrt{5}-\sqrt{2})$ $\displaystyle -\sqrt{6}(\sqrt{5}-\sqrt{2})$

Possible Answers:

$-\sqrt{30}-2\sqrt{3}$ $\displaystyle -\sqrt{30}-2\sqrt{3}$

$2\sqrt{3}-\sqrt{30}$ $\displaystyle 2\sqrt{3}-\sqrt{30}$

$12-\sqrt{30}$ $\displaystyle 12-\sqrt{30}$

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

$3\sqrt{2}-\sqrt{30}$ $\displaystyle 3\sqrt{2}-\sqrt{30}$

Correct answer:

$2\sqrt{3}-\sqrt{30}$ $\displaystyle 2\sqrt{3}-\sqrt{30}$

Explanation:

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

$\\-\sqrt{6}(\sqrt{5}-\sqrt{2})\\=-\sqrt{30}+\sqrt{12}\\=-\sqrt{30}+2\sqrt{3}$ $\displaystyle \\-\sqrt{6}(\sqrt{5}-\sqrt{2})\\=-\sqrt{30}+\sqrt{12}\\=-\sqrt{30}+2\sqrt{3}$

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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #4 : How To Multiply Square Roots

Example Question #2 : How To Multiply Square Roots

Example Question #1 : How To Multiply Square Roots

Example Question #6 : How To Multiply Square Roots

Example Question #7 : How To Multiply Square Roots

Example Question #2 : How To Multiply Square Roots

Example Question #1 : How To Multiply Square Roots

Example Question #81 : Arithmetic

Example Question #82 : Arithmetic

Example Question #83 : Arithmetic

All SAT Math Resources

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