SAT Math : How to find the common factor of square roots

Study concepts, example questions & explanations for SAT Math

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Example Questions

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

Solve for \dpi{100} x:

x\sqrt{45}+x\sqrt{72}=\sqrt{18}

Possible Answers:

x=\frac{\sqrt{2}}{\sqrt{5}}+\frac{1}{2}

x=3

x=\sqrt{9}

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

x=\frac{\sqrt{5}}{\sqrt{2}}+2

Correct answer:

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

Explanation:

x\sqrt{45}+x\sqrt{72}=\sqrt{18}

Notice how all of the quantities in square roots are divisible by 9

x\sqrt{9\times 5}+x\sqrt{9\times 8}=\sqrt{9\times 2}

x\sqrt{9}\sqrt{5}+x\sqrt{9}\sqrt{4\times 2}=\sqrt{9}\sqrt{2}

3x\sqrt{5}+3x\sqrt{4}\sqrt{2}=3\sqrt{2}

3x\sqrt{5}+6x\sqrt{2}=3\sqrt{2}

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

x=\frac{3\sqrt{2}}{3\sqrt{5}+6\sqrt{2}}

Simplifying, this becomes

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

Example Question #2 : How To Find The Common Factor Of Square Roots

Solve for :

Possible Answers:

Correct answer:

Explanation:

Note that all of the square root terms share a common factor of 36, which itself is a square of 6:

 

Factoring  from both terms on the left side of the equation:

 

 

Example Question #3 : How To Find The Common Factor Of Square Roots

Solve for :

Possible Answers:

Correct answer:

Explanation:

Note that both  and  have a common factor of  and  is a perfect square:

 

From here, we can factor  out of both terms on the lefthand side 

Example Question #4 : How To Find The Common Factor Of Square Roots

Solve for :

Possible Answers:

Correct answer:

Explanation:

In order to solve for , first note that all of the square root terms on the left side of the equation have a common factor of 9 and 9 is a perfect square:

 

Simplifying, this becomes:

Example Question #25 : Arithmetic

Which of the following is equivalent to:

?

Possible Answers:

Correct answer:

Explanation:

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

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

This is the same as:

These have a common .  Therefore, factor that out:

Example Question #26 : Arithmetic

Simplify:

Possible Answers:

Correct answer:

Explanation:

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

Now, this could be rewritten:

Now, note that 

Therefore, you can simplify again:

Now, that looks messy! Still, if you look carefully, you see that all of your factors have ; therefore, factor that out:

This is the same as:

Example Question #5 : How To Find The Common Factor Of Square Roots

Solve for :

Possible Answers:

Correct answer:

Explanation:

Examining the terms underneath the radicals, we find that  and  have a common factor of  itself is a perfect square, being the product of  and . Hence, we recognize that the radicals can be re-written in the following manner:

, and .

The equation can then be expressed in terms of these factored radicals as shown:

 

  

Factoring the common term  from the lefthand side of this equation yields

Divide both sides by the expression in the parentheses:

Divide both sides by  to yield  by itself on the lefthand side:

Simplify the fraction on the righthand side by dividing the numerator and denominator by :

This is the solution for the unknown variable  that we have been required to find.

 

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