All ACT Math Resources
Example Questions
Example Question #1 : Factoring Common Factors Of Squares And Square Roots
To solve the equation , we can first factor the numbers under the square roots.
When a factor appears twice, we can take it out of the square root.
Now the numbers can be added directly because the expressions under the square roots match.
Example Question #2 : Factoring Common Factors Of Squares And Square Roots
Solve for .
First, we can simplify the radicals by factoring.
Now, we can factor out the .
Now divide and simplify.
Example Question #2 : Factoring Common Factors Of Squares And Square Roots
Which of the following is equivalent to:
?
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 #3 : Factoring Common Factors Of Squares And Square Roots
Simplify:
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 #1 : Factoring Common Factors Of Squares And Square Roots
Simplify:
Begin by factoring out the relevant squared data:
is the same as
This can be simplified to:
Since your various factors contain square roots of , you can simplify:
Technically, you can factor out a :
Example Question #2 : How To Find The Common Factor Of Square Roots
Solve for :
Begin by breaking apart the square roots on the left side of the equation:
This can be rewritten:
You can combine like terms on the left side:
Solve by dividing both sides by :
This simplifies to:
Example Question #6 : Basic Squaring / Square Roots
Solve for :
To begin solving this problem, find the greatest common perfect square for all quantities under a radical.
--->
Pull out of each term on the left:
--->
Next, factor out from the left-hand side:
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Lastly, isolate :
--->
Example Question #7 : Basic Squaring / Square Roots
Solve for :
Solving this one is tricky. At first glance, we have no common perfect square to work with. But since each term can produce the quantity , let's start there:
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Simplify the first term:
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Divide all terms by to simplify,
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Next, factor out from the left-hand side:
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Isolate by dividing by and simplifying:
--->
Last, simplify the denominator:
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Example Question #34 : Arithmetic
Solve for :
Right away, we notice that is a prime radical, so no simplification is possible. Note, however, that both other radicals are divisible by .
Our first step then becomes simplifying the equation by dividing everything by :
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Next, factor out from the left-hand side:
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Lastly, isolate :
--->
Example Question #8 : How To Find The Common Factor Of Square Roots
Solve for :
Once again, there are no common perfect squares under the radical, but with some simplification, the equation can still be solved for :
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Simplify:
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Factor out from the left-hand side:
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Lastly, isolate :
--->