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PSAT Math : Arithmetic

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Example Questions

Example Question #3 : Basic Squaring / Square Roots

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

Possible Answers:

$5\sqrt{7}$

$\sqrt{5}$

$\sqrt{7}$

$3\sqrt{11}$

$\sqrt{35}$

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 #4 : Basic Squaring / Square Roots

Simplify.

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

Possible Answers:

$12\sqrt{2}+8$

$28\sqrt{2}$

$16\sqrt{2}+8$

$6\sqrt{2}+8$

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 #181 : Arithmetic

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

Possible Answers:

$9$

$81$

$729$

$27$

$3$

Correct answer:

$81$

Explanation:

Square both sides:

x = (3²)²= 9² = 81

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Example Question #7 : Basic Squaring / Square Roots

Simplify:

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

Possible Answers:

$-\sqrt{3}+9$

$7\sqrt{3}+9$

$2\sqrt{3}$

$\sqrt{63}$

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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Example Question #1 : How To Add Square Roots

Simplify the expression:

$\sqrt{45} + \sqrt{125} + \sqrt{175}$

Possible Answers:

The expression cannot be siimplified further.

$5 \sqrt{3} + 5 \sqrt{ 5}+5 \sqrt{ 7}$

$15 \sqrt{5}$

$5 \sqrt{3} + 12 \sqrt{ 5}$

$8 \sqrt{ 5}+5 \sqrt{ 7}$

Correct answer:

$8 \sqrt{ 5}+5 \sqrt{ 7}$

Explanation:

For each of the expressions, factor out a perfect square:

$\sqrt{45} + \sqrt{125} + \sqrt{175}$

$= \sqrt{9 \cdot 5} + \sqrt{25 \cdot 5} + \sqrt{25 \cdot 7}$

$= \sqrt{9 } \cdot \sqrt{ 5} + \sqrt{25 } \cdot \sqrt{ 5}+ \sqrt{25} \cdot \sqrt{ 7}$

$=3 \sqrt{ 5} +5 \sqrt{ 5}+5 \sqrt{ 7}$

$=\left (3 +5 \right ) \sqrt{ 5}+5 \sqrt{ 7}$

$=8 \sqrt{ 5}+5 \sqrt{ 7}$

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Example Question #2 : How To Add Square Roots

Simplify:

$\sqrt{54} - \sqrt{24} + \sqrt{6}$

Possible Answers:

$2 \sqrt{6}$

The expression cannot be simplified further.

$4\sqrt{6}$

$6 \sqrt{6}$

Correct answer:

$2 \sqrt{6}$

Explanation:

Simplify each of the radicals by factoring out a perfect square:

$\sqrt{54} - \sqrt{24} + \sqrt{6}$

$= \sqrt{9 \cdot 6} - \sqrt{4\cdot 6} + \sqrt{6}$

$= \sqrt{9} \cdot \sqrt{6} - \sqrt{4} \cdot \sqrt{6} + \sqrt{6}$

$= 3\sqrt{6} - 2 \sqrt{6} + 1\sqrt{6}$

$=\left ( 3 - 2 + 1 \right ) \sqrt{6}$

$= 2 \sqrt{6}$

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Example Question #3 : How To Add Square Roots

Add the square roots into one term:

$\small {\sqrt{12}} + {\sqrt{75}}$

Possible Answers:

None of the other answers

$\small 5{\sqrt{3}}$

$\small 6{\sqrt{3}}$

$\small 7{\sqrt{3}}$

$\small \small 8{\sqrt{3}}$

Correct answer:

$\small 7{\sqrt{3}}$

Explanation:

In order to solve this problem we need to simplfy each of the radicals. By doing this we will get two terms that have the same number under the radical which will allow us to combine the terms.

$\small {\sqrt{12}} + {\sqrt{75}}$

$\small = \small {\sqrt{4}} \times {\sqrt{3}} + {\sqrt{25}} \times {\sqrt{3}}$

$\small \small = 2 {\sqrt{3}} + 5 {\sqrt{3}}$

$\small = 7 {\sqrt{3}}$

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Example Question #3 : How To Add Square Roots

Simplify:

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

Possible Answers:

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

$6\sqrt{10}$

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

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

$6\sqrt{30}$

Correct answer:

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

Explanation:

Remember that you treat square roots like you do variables in the sense that you just add the like factors. In this problem, the only set of like factors is the pair of $\sqrt{2}$ values. Hence:

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

Do not try to simplify any further!

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Example Question #4 : How To Add Square Roots

Simplify:

$\sqrt{2}+3\sqrt{12}+2\sqrt{50}+\sqrt{3}$

Possible Answers:

$18\sqrt{5}$

$\sqrt{2}+4\sqrt{3}+3\sqrt{12}$

$11\sqrt{2}+7\sqrt{3}$

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

$18\sqrt{6}$

Correct answer:

$11\sqrt{2}+7\sqrt{3}$

Explanation:

Begin by simplifying your more complex roots:

$\sqrt{12}= \sqrt{2*2*3}=2\sqrt{3}$

$\sqrt{50} = \sqrt{5*5*2}=5\sqrt{2}$

This lets us rewrite our expression:

$\sqrt{2}+3\sqrt{12}+2\sqrt{50}+\sqrt{3} = \sqrt{2}+3(2\sqrt{3})+2(5\sqrt{2})+\sqrt{3}$

Do the basic multiplications of coefficients:

$\sqrt{2}+3(2\sqrt{3})+2(5\sqrt{2})+\sqrt{3} = \sqrt{2}+6\sqrt{3}+10\sqrt{2}+\sqrt{3}$

Reorder the terms:

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

Finally, combine like terms:

$11\sqrt{2}+7\sqrt{3}$

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Example Question #1 : How To Multiply Square Roots

Multiply and simplify. Assuming all integers are positive real numbers.

$2\sqrt{5}\cdot6\sqrt{20}$

Possible Answers:

120

160

Correct answer:

120

Explanation:

$2\sqrt{5}\cdot6\sqrt{20}$

Multiply the coefficents outside of the radicals.

$2\cdot6= 12$

Then multiply the radicans. Simplify by checking for a perfect square.

$\sqrt{5\cdot 20}=\sqrt{100}=10$

Final answer is your leading coefficent, , multiplied by the answer acquired by multiplying the terms under the radican, .

$12\cdot10= 120$

The final answer is 120 .

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PSAT Math : Arithmetic

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #3 : Basic Squaring / Square Roots

Example Question #4 : Basic Squaring / Square Roots

Example Question #181 : Arithmetic

Example Question #7 : Basic Squaring / Square Roots

Example Question #1 : How To Add Square Roots

Example Question #2 : How To Add Square Roots

Example Question #3 : How To Add Square Roots

Example Question #3 : How To Add Square Roots

Example Question #4 : How To Add Square Roots

Example Question #1 : How To Multiply Square Roots

All PSAT Math Resources

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