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PSAT Math : Basic Squaring / Square Roots

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Example Question #1 : Square Roots And Operations

Divide and simplify. Assume all integers are positive real numbers.

$\sqrt{\frac{32}{4}}$

Possible Answers:

$\sqrt{8}$

$\sqrt{2}$

$2\sqrt{2}$

$4\sqrt{2}$

Correct answer:

$2\sqrt{2}$

Explanation:

$\sqrt{\frac{32}{4}}$

There are two ways to solve this problem. First you can divide the numbers under the radical. Then simplify.

Example 1

$\sqrt{\frac{32}{4}}=\sqrt{8}=\sqrt{4}\cdot\sqrt{2}=2\sqrt{2}$

Example 2

Find the square root of both numerator and denominator, simplifying as much as possible then dividing out like terms.

$\sqrt{\frac{32}{4}}=\frac{\sqrt{32}}{\sqrt{4}}=\frac{\sqrt{16}\cdot \sqrt{2}}{\sqrt{4}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$

Both methods will give you the correct answer of $2\sqrt{2}$ .

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Example Question #1 : How To Divide Square Roots

(√27 + √12) / √3 is equal to

Possible Answers:

√3

(6√3)/√3

5/√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 #1 : How To Divide Square Roots

Simplify:

$\sqrt{\frac{64}{169}}$

Possible Answers:

$\frac{8}{15}$

$\frac{8}{13}$

$\frac{64}{169}$

$\frac{7}{13}$

Correct answer:

$\frac{8}{13}$

Explanation:

To simplfy, we must first distribute the square root.

$\frac{\sqrt{64}}{\sqrt{169}}$

Next, we can simplify each of the square roots.

$\frac{8}{13}$

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Example Question #3 : How To Divide Square Roots

Find the quotient:

$\sqrt{\frac{18}{2}}$

Possible Answers:

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

$\sqrt{2}$

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

Correct answer:

Explanation:

Find the quotient:

$\frac{\sqrt{18}}{\sqrt{2}}$

There are two ways to approach this problem.

Option 1: Combine the radicals first, the reduce

$\frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}}=\sqrt{9}=3$

Option 2: Simplify the radicals first, then reduce

$\frac{\sqrt{18}}{\sqrt{2}}=\frac{\sqrt{9\cdot 2}}{\sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{2}}=3$

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Example Question #4 : How To Divide Square Roots

Find the quotient:

$\frac{\sqrt{54}}{\sqrt{45}}$

Possible Answers:

$3\sqrt{6}$

$\frac{\sqrt{30}}{5}$

$9\sqrt{5}$

$\frac{\sqrt{}6}{\sqrt{}5}$

Correct answer:

$\frac{\sqrt{30}}{5}$

Explanation:

Simplify each radical:

$\frac{\sqrt{54}}{\sqrt{45}} = \frac{\sqrt{9\cdot 6}}{\sqrt{9\cdot 5}}=\frac{3\sqrt{6}}{3\sqrt{5}}=\frac{\sqrt{6}}{\sqrt{5}}$

Rationalize the denominator:

$\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}= \frac{\sqrt{30}}{5}$

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Example Question #21 : Square Roots And Operations

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

Possible Answers:

$6\sqrt{3}-9$

$24\sqrt{3}-9$

$24\sqrt{3}-81$

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

None of the available answers

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 : How To Subtract Square Roots

$\sqrt{63}-\sqrt{28}$

Possible Answers:

$\sqrt{5}$

$\sqrt{7}$

$\sqrt{35}$

$3\sqrt{11}$

$5\sqrt{7}$

Correct answer:

$\sqrt{7}$

Explanation:

Step one: Find the greatest square factor of each radical

For this is , and for it is .

Therefore:

$\sqrt{63}-\sqrt{28}=\sqrt{9*7}-\sqrt{4*7}$

Step two: Simplify the radicals

$\sqrt{9*7}-\sqrt{4*7}$

$3\sqrt{7}-2\sqrt{7}$

$\sqrt{7}$

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Example Question #1 : How To Add Square Roots

Simplify.

$\sqrt{32}+\sqrt{64}+\sqrt{8}$

Possible Answers:

$28\sqrt{2}$

$12\sqrt{2}+8$

$16\sqrt{2}+8$

$6\sqrt{2}+8$

Correct answer:

$6\sqrt{2}+8$

Explanation:

$\sqrt{32}+\sqrt{64}+\sqrt{8}$

First step is to find perfect squares in all of our radicans.

$\sqrt{32}\rightarrow\sqrt{16\cdot 2}\rightarrow4\sqrt{2}$

$\sqrt{64}\rightarrow\sqrt{8\cdot 8}\rightarrow8$

$\sqrt{8}\rightarrow\sqrt{4\cdot 2}\rightarrow2\sqrt{2}$

After doing so you are left with $4\sqrt{2}+2\sqrt{2}+8$

*Just like fractions you can only add together coefficents with like terms under the radical. *

$6\sqrt{2}+8$

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Example Question #1 : How To Add Square Roots

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

Possible Answers:

$3$

$81$

$9$

$27$

$729$

Correct answer:

$81$

Explanation:

Square both sides:

x = (3²)²= 9² = 81

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Example Question #1 : Basic Squaring / Square Roots

Simplify:

$\sqrt{27}-\sqrt{48}+\sqrt{81}$

Possible Answers:

$2\sqrt{3}$

$-\sqrt{3}+9$

$7\sqrt{3}+9$

$\sqrt{63}$

Correct answer:

$-\sqrt{3}+9$

Explanation:

To combine radicals, they must have the same radicand. Therefore, we must find the perfect squares in each of our square roots and pull them out.

$\sqrt{27}\rightarrow \sqrt{9*3}\rightarrow \sqrt{3^2*3}\rightarrow 3\sqrt{3}$

$\sqrt{48}\rightarrow \sqrt{16*3}\rightarrow \sqrt{4^2*3}\rightarrow 4\sqrt{3}$

$\sqrt{81}\rightarrow\sqrt{9*9}\rightarrow\sqrt{9^2}\rightarrow9$

Now, we plug these equivalent expressions back into our equation and simplify:

$\sqrt{27}-\sqrt{48}+\sqrt{81}$

$3\sqrt3-4\sqrt3+9$

$-\sqrt3+9$

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PSAT Math : Basic Squaring / Square Roots

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Square Roots And Operations

Example Question #1 : How To Divide Square Roots

Example Question #1 : How To Divide Square Roots

Example Question #3 : How To Divide Square Roots

Example Question #4 : How To Divide Square Roots

Example Question #21 : Square Roots And Operations

Example Question #1 : How To Subtract Square Roots

Example Question #1 : How To Add Square Roots

Example Question #1 : How To Add Square Roots

Example Question #1 : Basic Squaring / Square Roots

All PSAT Math Resources

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