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 :Factoring And Simplifying Square Roots

Solve for\dpi{100} x:

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

Possible Answers:

x=3

x=\sqrt{9}

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

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

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

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

3.x\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 #1 :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:

Factoringfrom both terms on the left side of the equation:

Example Question #2 :Basic Squaring / Square Roots

Solve for:

Possible Answers:

Correct answer:

Explanation:

Note that bothandhave a common factor ofandis a perfect square:

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

Example Question #3 :Basic Squaring / 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 #2 :How To Find The Common Factor Of Square Roots

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

Simplify:

Possible Answers:

Correct answer:

Explanation:

These three roots all have ain 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 #4 :Basic Squaring / Square Roots

Solve for:

Possible Answers:

Correct answer:

Explanation:

Examining the terms underneath the radicals, we find thatandhave a common factor of.itself is a perfect square, being the product ofand. 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 termfrom the lefthand side of this equation yields

Divide both sides by the expression in the parentheses:

Divide both sides byto yieldby 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 variablethat we have been required to find.

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