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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #21 : Basic Squaring / Square Roots

Simplify:

$-2\sqrt{5}(7\sqrt{3}+3\sqrt{10})$

Possible Answers:

$-19\sqrt{10}$

$-14\sqrt{15}-30\sqrt{2}$

$-14\sqrt{5}+30\sqrt{2}$

$-14\sqrt{5}-5\sqrt{2}$

$14\sqrt{5}-6\sqrt{2}$

Correct answer:

$-14\sqrt{15}-30\sqrt{2}$

Explanation:

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

$\\-2\sqrt{5}(7\sqrt{3}+3\sqrt{10})\\=-14\sqrt{15}-6\sqrt{50}$

$\\-14\sqrt{15}-6\sqrt{50}\\=-14\sqrt{15}-6*5\sqrt{2}\\=-14\sqrt{15}-30\sqrt{2}$

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Example Question #21 : Basic Squaring / Square Roots

Simplify:

$(\sqrt{2}+\sqrt{7})(\sqrt{27}+\sqrt{12})$

Possible Answers:

$5\sqrt{6}+5\sqrt{21}$

$3\sqrt{6}+2\sqrt{2}+2\sqrt{3}+3\sqrt{7}$

$5\sqrt{6}+2\sqrt{21}+3\sqrt{2}$

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

$3\sqrt{3}+2\sqrt{21}$

Correct answer:

$5\sqrt{6}+5\sqrt{21}$

Explanation:

$(\sqrt{2}+\sqrt{7})(\sqrt{27}+\sqrt{12})$

Let's simplify the right parentheses.

$(\sqrt{27}+\sqrt{12})=(3\sqrt{{3}}+2\sqrt{3})=5\sqrt{3}$

Now we can distribute the radical to each term in the parentheses.

$5\sqrt{3}(\sqrt{2}+\sqrt{7})=5\sqrt{6}+5\sqrt{21}$

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Example Question #1 : Basic Squaring / Square Roots

If $\sqrt{x}=3^2$ what is $x$ ?

Possible Answers:

$27$

$9$

$3$

$81$

$729$

Correct answer:

$81$

Explanation:

Square both sides:

x = (3²)²= 9² = 81

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Example Question #22 : Basic Squaring / Square Roots

Simplify in radical form:

$\sqrt{507}+\sqrt{12}$

Possible Answers:

$\sqrt{519}$

$26\sqrt{3}$

$15\sqrt{3}$

$11\sqrt{6}$

$12\sqrt{6}$

Correct answer:

$15\sqrt{3}$

Explanation:

To simplify, break down each square root into its component factors:

$\sqrt{507}+\sqrt{12}$

$\sqrt{169\cdot 3}+\sqrt{4\cdot 3}$

$\sqrt{13\cdot13\cdot3}\:+\sqrt{2\cdot 2\cdot 3}$

You can remove pairs of factors and bring them outside the square root sign. At this point, since each term shares $\sqrt3$ , you can add them together to yield the final answer:

$13\sqrt{3}+2\sqrt{3}=15\sqrt{3}$

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Example Question #1 : How To Add Square Roots

Simplify: $\frac{2\sqrt{3}}{\sqrt{2}}+\frac{4\sqrt{2}}{\sqrt{3}}$

Possible Answers:

$2\sqrt{6}-4\sqrt{2}$

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

$2\sqrt{6}$

$\frac{7\sqrt{6}}{3}$

None of the other answers

Correct answer:

$\frac{7\sqrt{6}}{3}$

Explanation:

Take each fraction separately first:

(2√3)/(√2) = [(2√3)/(√2)] * [(√2)/(√2)] = (2 * √3 * √2)/(√2 * √2) = (2 * √6)/2 = √6

Similarly:

(4√2)/(√3) = [(4√2)/(√3)] * [(√3)/(√3)] = (4√6)/3 = (4/3)√6

Now, add them together:

√6 + (4/3)√6 = (3/3)√6 + (4/3)√6 = (7/3)√6

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Example Question #11 : Square Roots And Operations

Simplify the following expression: $\sqrt{60}+\sqrt{40}+\sqrt{10}$

Possible Answers:

$2\sqrt{15}+3\sqrt{10}$

$4\sqrt{15}+5\sqrt{5}$

$5\sqrt{25}$

$2\sqrt{15}+2\sqrt{20}$

Correct answer:

$2\sqrt{15}+3\sqrt{10}$

Explanation:

Begin by factoring out each of the radicals:

$\sqrt{60}+\sqrt{40}+\sqrt{10}=\sqrt{4*15}+\sqrt{4*10}+\sqrt{10}$

For the first two radicals, you can factor out a $\sqrt{4}$ or :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}$

The other root values cannot be simply broken down. Now, combine the factors with $\sqrt{10}$ :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}=2\sqrt{15}+3\sqrt{10}$

This is your simplest form.

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Example Question #4 : Square Roots And Operations

Solve for .

Note, $x\ge0$ :

$15\sqrt{x}-10=4\sqrt{x}+4$

Possible Answers:

$\frac{841}{14}$

$\frac{196}{121}$

$\frac{841}{5}$

$\frac{196}{11}$

Correct answer:

$\frac{196}{121}$

Explanation:

Begin by getting your terms onto the left side of the equation and your numeric values onto the right side of the equation:

$15\sqrt{x}-4\sqrt{x}=4+10$

Next, you can combine your radicals. You do this merely by subtracting their respective coefficients:

$11\sqrt{x}=14$

Now, square both sides:

$(11\sqrt{x})^2=(14)^2$

$(11)^2(\sqrt{x})^2=(14)^2$

121x=196

Solve by dividing both sides by 121 :

$x=\frac{196}{121}$

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Example Question #21 : Square Roots And Operations

Evaluate: $\sqrt{108}-\sqrt{81}$

Possible Answers:

$6\sqrt{3}-9$

None of the available answers

$24\sqrt{3}-9$

$4\sqrt{27}-\sqrt{9}$

$24\sqrt{3}-81$

Correct answer:

$6\sqrt{3}-9$

Explanation:

Let us factor 108 and 81

$108=2\times 54= 2\times 2\times 27= 2^2\times 3\times 9= 2^2\times 3^2\times 3$

$81=9\times 9=9^2$

$\sqrt{108}-\sqrt{81}=6\sqrt{3}-9$

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Example Question #1 : Basic Squaring / Square Roots

(√27 + √12) / √3 is equal to

Possible Answers:

5/√3

√3

(6√3)/√3

Correct answer:

Explanation:

√27 is the same as 3√3, while √12 is the same as 2√3.

3√3 + 2√3 = 5√3

(5√3)/(√3) = 5

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Example Question #91 : Arithmetic

Simplify: $\frac{\sqrt{210}}{\sqrt{6}}$

Possible Answers:

$7\sqrt{5}$

$\sqrt{35}$

$5\sqrt{7}$

$\sqrt{13}$

Correct answer:

$\sqrt{35}$

Explanation:

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

$\frac{\sqrt{210}}{\sqrt{6}}=\frac{\sqrt{35}*\sqrt{6}}{\sqrt{6}}=\sqrt{35}$

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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #21 : Basic Squaring / Square Roots

Example Question #21 : Basic Squaring / Square Roots

Example Question #1 : Basic Squaring / Square Roots

Example Question #22 : Basic Squaring / Square Roots

Example Question #1 : How To Add Square Roots

Example Question #11 : Square Roots And Operations

Example Question #4 : Square Roots And Operations

Example Question #21 : Square Roots And Operations

Example Question #1 : Basic Squaring / Square Roots

Example Question #91 : Arithmetic

All SAT Math Resources

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