ACT Math : Factoring Common Factors of Squares and Square Roots

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #31 : Arithmetic

Solve for :

Possible Answers:

Correct answer:

Explanation:

To begin solving this problem, find the greatest perfect square for all quantities under a radical.  might seem intimidating, but remember that raising even single-digit numbers to the fourth power creates huge numbers. In this case,  is divisible by , a perfect fourth power.

 ---> 

Pull the perfect terms out of each term on the left:

 ---> 

Next, factor out  from the left-hand side:

 ---> 

Lastly, isolate , remembering to simplify the fraction where possible:

 ---> 

Example Question #11 : Basic Squaring / Square Roots

Simplify: 

Possible Answers:

Correct answer:

Explanation:

To start, begin pulling the largest perfect square you can out of each number:

So, . You can just add the two terms together once they have a common radical.

Example Question #11 : Basic Squaring / Square Roots

Simplify: 

Possible Answers:

Correct answer:

Explanation:

Again here, it is easiest to recognize that both of our terms are divisible by , a prime number likely to appear in our final answer:

Now, simplify our perfect squares:

Lastly, subtract our terms with a common radical:

Example Question #11 : Basic Squaring / Square Roots

Solve for 

Possible Answers:

Correct answer:

Explanation:

To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

 ---> 

Factor out the square root of each perfect square:

 ---> 

Next, factor out  from each term on the left-hand side of the equation:

 ---> 

Lastly, isolate :

 ---> 

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