GRE Math : Factoring Common Factors of Squares and Square Roots

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Example Questions

Example Question #21 : Arithmetic

Which of the following is equivalent to:

$\sqrt{210}+\sqrt{55}$ ?

Possible Answers:

$5\sqrt{7}+\sqrt{11}$

$7\sqrt{30}+5\sqrt{11}$

$5\sqrt{462}$

$\sqrt{5}(\sqrt{42}+\sqrt{11})$

$\sqrt{265}$

Correct answer:

$\sqrt{5}(\sqrt{42}+\sqrt{11})$

Explanation:

To begin with, factor out the contents of the radicals. This will make answering much easier:

$\sqrt{210}+\sqrt{55}=\sqrt{2*3*5*7}+\sqrt{5*11}$

They both have a common factor . This means that you could rewrite your equation like this:

$\sqrt{5}\sqrt{2*3*7}+\sqrt{5}\sqrt{11}$

This is the same as:

$\sqrt{5}\sqrt{42}+\sqrt{5}\sqrt{11}$

These have a common $\sqrt{5}$ . Therefore, factor that out:

$\sqrt{5}\sqrt{42}+\sqrt{5}\sqrt{11}=\sqrt{5}(\sqrt{42}+\sqrt{11})$

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Example Question #1 : How To Find The Common Factor Of Square Roots

Simplify:

$\sqrt{15}-\sqrt{20}+\sqrt{35}$

Possible Answers:

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

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

$\sqrt{2}(\sqrt{5}+2\sqrt{7})$

$\sqrt{5}(\sqrt{10}-2)$

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

Correct answer:

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

Explanation:

These three roots all have a in common; therefore, you can rewrite them:

$\sqrt{15}-\sqrt{20}+\sqrt{35}=\sqrt{3*5}-\sqrt{4*5}+\sqrt{7*5}$

Now, this could be rewritten:

$\sqrt{3*5}-\sqrt{4*5}+\sqrt{7*5}=\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{4}+\sqrt{5}\sqrt{7}$

Now, note that $\sqrt{4}=2$

Therefore, you can simplify again:

$\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{4}+\sqrt{5}\sqrt{7}=\sqrt{5}\sqrt{3}-2\sqrt{5}+\sqrt{5}\sqrt{7}$

Now, that looks messy! Still, if you look carefully, you see that all of your factors have $\sqrt{5}$ ; therefore, factor that out:

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

This is the same as:

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

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Example Question #31 : Basic Squaring / Square Roots

Simplify the following:

$\sqrt{125}+\sqrt{245}+\sqrt{80}$

Possible Answers:

$90\sqrt{5}$

It cannot be simplified any further

$\sqrt{450}$

$16\sqrt{5}$

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

Correct answer:

$16\sqrt{5}$

Explanation:

Begin by factoring each of the roots to see what can be taken out of each:

$\sqrt{5^3}+\sqrt{7^2*5}+\sqrt{4^2*5}$

These can be rewritten as:

$5\sqrt{5}+7\sqrt{5}+4\sqrt{5}$

Notice that each of these has a common factor of $\sqrt{5}$ . Thus, we know that we can rewrite it as:

$5\sqrt{5}+7\sqrt{5}+4\sqrt{5} = \sqrt{5}(5+7+4) = 16\sqrt{5}$

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Example Question #1 : How To Find The Common Factor Of Square Roots

Simplify the following:

$\sqrt{40}+\sqrt{20}+\sqrt{160}$

Possible Answers:

$8\sqrt{10}$

$\sqrt{10}(6+\sqrt{2})$

The expression cannot be simplified any further.

$4\sqrt{20}$

$\sqrt{5}(5 + 2\sqrt{2})$

Correct answer:

$\sqrt{10}(6+\sqrt{2})$

Explanation:

Clearly, all three of these roots have a common factor inside of their radicals. We can start here with our simplification. Therefore, rewrite the radicals like this:

$\sqrt{4}\sqrt{10}+\sqrt{2}\sqrt{10}+\sqrt{16}\sqrt{10}$

We can simplify this a bit further:

$2\sqrt{10}+\sqrt{2}\sqrt{10}+4\sqrt{10} = 6\sqrt{10} + \sqrt{2}\sqrt{10}$

From this, we can factor out the common $\sqrt{10}$ :

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

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Example Question #31 : Arithmetic

$\frac{\sqrt{243}}{\sqrt{48}}=$

Possible Answers:

$4\sqrt{3}$

$9\sqrt{3}$

$\frac{81}{16}$

$\frac{3}{2}$

$\frac{9}{4}$

Correct answer:

$\frac{9}{4}$

Explanation:

To attempt this problem, attempt to simplify the roots of the numerator and denominator:

$\frac{\sqrt{243}}{\sqrt{48}}=\frac{\sqrt{3*81}}{\sqrt{3*16}}$

Notice how both numerator and denominator have a perfect square:

$\frac{\sqrt{243}}{\sqrt{48}}=\frac{\sqrt{3*81}}{\sqrt{3*16}}=\frac{9\sqrt3}{4\sqrt3}$

The $\sqrt3$ term can be eliminated from the numerator and denominator, leaving

$\frac{9}{4}$

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Example Question #1 : Factoring Common Factors Of Squares And Square Roots

$\frac{\sqrt{343}}{\sqrt{63}}=$

Possible Answers:

$7\sqrt{7}$

$\frac{3}{7}$

$\frac{49}{9}$

$3\sqrt{3}$

$\frac{7}{3}$

Correct answer:

$\frac{7}{3}$

Explanation:

For this problem, begin by simplifying the roots. As it stands, numerator and denominator have a common factor of in the radical:

$\frac{\sqrt{343}}{\sqrt{63}}=\frac{\sqrt{7*49}}{\sqrt{7*9}}$

And as it stands, this is multiplied by a perfect square in the numerator and denominator:

$\frac{\sqrt{343}}{\sqrt{63}}=\frac{\sqrt{7*49}}{\sqrt{7*9}}=\frac{7\sqrt{7}}{3\sqrt{7}}$

The $\sqrt{7}$ term can be eliminated from the top and bottom, leaving

$\frac{7}{3}$

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Example Question #1 : Factoring Common Factors Of Squares And Square Roots

$\frac{\sqrt{150}}{\sqrt{48}}=$

Possible Answers:

$\frac{25\sqrt{2}}{16}$

$5\sqrt{2}$

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

$\frac{5\sqrt{2}}{4}$

$\frac{5\sqrt{3}}{4}$

Correct answer:

$\frac{5\sqrt{2}}{4}$

Explanation:

To solve this problem, try simplifying the roots by factoring terms; it may be noticeable from observation that both numerator and denominator have a factor of in the radical:

$\frac{\sqrt{150}}{\sqrt{48}}=\frac{\sqrt{3*50}}{\sqrt{3*16}}$

We can see that the denominator has a perfect square; now try factoring the in the numerator:

$\frac{\sqrt{150}}{\sqrt{48}}=\frac{\sqrt{3*50}}{\sqrt{3*16}}=\frac{\sqrt{3*2*25}}{4\sqrt{3}}$

We can see that there's a perfect square in the numerator:

$\frac{\sqrt{150}}{\sqrt{48}}=\frac{\sqrt{3*50}}{\sqrt{3*16}}=\frac{\sqrt{3*2*25}}{4\sqrt{3}}=\frac{5\sqrt{3*2}}{4\sqrt{3}}$

Since there is a in the radical in both the numerator and denominator, we can eliminate it, leaving

$\frac{5\sqrt{2}}{4}$

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GRE Math : Factoring Common Factors of Squares and Square Roots

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #21 : Arithmetic

Example Question #1 : How To Find The Common Factor Of Square Roots

Example Question #31 : Basic Squaring / Square Roots

Example Question #1 : How To Find The Common Factor Of Square Roots

Example Question #31 : Arithmetic

Example Question #1 : Factoring Common Factors Of Squares And Square Roots

Example Question #1 : Factoring Common Factors Of Squares And Square Roots

All GRE Math Resources

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