How to add square roots

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PSAT Math › How to add square roots

Questions 1 - 8

1

Simplify:

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

$-\sqrt{3}+9$

$\sqrt{63}$

$2\sqrt{3}$

$7\sqrt{3}+9$

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:

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

$3\sqrt3-4\sqrt3+9$

$-\sqrt3+9$

2

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

$81$

$9$

$3$

$729$

$27$

Explanation

Square both sides:

x = (32)2 = 92 = 81

3

Simplify:

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

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

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

$6\sqrt{10}$

$6\sqrt{30}$

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

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!

4

Simplify:

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

$2 \sqrt{6}$

$6 \sqrt{6}$

$4\sqrt{6}$

The expression cannot be simplified further.

Explanation

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

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

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

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

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

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

$= 2 \sqrt{6}$

5

Simplify the expression:

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

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

$15 \sqrt{5}$

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

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

The expression cannot be siimplified further.

Explanation

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

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

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

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

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

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

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

6

Simplify.

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

$6\sqrt{2}+8$

$16\sqrt{2}+8$

$28\sqrt{2}$

$12\sqrt{2}+8$

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. *

$6\sqrt{2}+8$

7

Add the square roots into one term:

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

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

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

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

None of the other answers

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

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.

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

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

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

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

8

Simplify:

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

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

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

$18\sqrt{6}$

$18\sqrt{5}$

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

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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