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PSAT Math : How to add square roots

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Example Questions

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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Example Question #181 : Arithmetic

Simplify the expression:

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

Possible Answers:

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

The expression cannot be siimplified further.

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

$15 \sqrt{5}$

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

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 #1 : How To Add Square Roots

Simplify:

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

Possible Answers:

The expression cannot be simplified further.

$4\sqrt{6}$

$2 \sqrt{6}$

$6 \sqrt{6}$

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 #1 : How To Add Square Roots

Add the square roots into one term:

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

Possible Answers:

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

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

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

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

None of the other answers

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 #1 : How To Add Square Roots

Simplify:

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

Possible Answers:

$6\sqrt{10}$

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

$6\sqrt{30}$

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

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

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 #11 : Square Roots And Operations

Simplify:

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

Possible Answers:

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

$18\sqrt{5}$

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

$18\sqrt{6}$

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

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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PSAT Math : How to add square roots

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Add Square Roots

Example Question #1 : How To Add Square Roots

Example Question #1 : Basic Squaring / Square Roots

Example Question #181 : Arithmetic

Example Question #1 : How To Add Square Roots

Example Question #1 : How To Add Square Roots

Example Question #1 : How To Add Square Roots

Example Question #11 : Square Roots And Operations

All PSAT Math Resources

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