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ACT Math : Factoring Squares

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Example Questions

Example Question #1 : Factoring Squares

Which real number satisfies $3^{x}*9=27^{2}$ ?

Possible Answers:

Correct answer:

Explanation:

Simplify the base of 9 and 27 in order to have a common base.

(3^x)(9)=27²

= (3^x)(3²)=(3³)²

=(3^x+2)=3⁶

Therefore:

x+2=6

x=4

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Example Question #2 : How To Factor A Common Factor Out Of Squares

Which of the following is a factor of $4x^{4}+ 36 x^{2}$ ?

Possible Answers:

$x^{2}+ 9$

$x^{2}+ 3$

$2x^{2}+ 3$

$x^{2}+ 6$

$2x^{2}+ 9$

Correct answer:

$x^{2}+ 9$

Explanation:

The terms of $4x^{4}+ 36 x^{2}$ have $4x^{2}$ as their greatest common factor, so

$4x^{4}+ 36 x^{2} = 4x^{2} (x^{2}+9)$

$x^{2}+ 9$ is a prime polynomial.

Of the five choices, only $x^{2}+ 9$ is a factor.

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Example Question #2 : How To Factor A Common Factor Out Of Squares

Simplify $\frac{8^{3} \times 3}{2^{6}\times 27}$

Possible Answers:

$\frac{9}{8}$

$\frac{4}{3}$

$\frac{16}{9}$

$\frac{32}{9}$

$\frac{8}{9}$

Correct answer:

$\frac{8}{9}$

Explanation:

The easiest way to approach this problem is to break everything into exponents. $8^{3}$ is equal to $2^{9}$ and 27 is equal to $3^{3}$ . Therefore, the expression can be broken down into $\frac{2^{9} \times 3}{2^{6}\times 3^{3}}$ . When you cancel out all the terms, you get $\frac{2^{3}}{3^{2}}$ , which equals $\frac{8}{9}$ .

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Example Question #4 : Squaring / Square Roots / Radicals

Which of the following expression is equal to

$\sqrt{(45)(9)+(27)(36)}$

Possible Answers:

$7\sqrt{15}$

$9\sqrt{17}$

$9\sqrt{15}$

$7\sqrt{17}$

$9\sqrt{19}$

Correct answer:

$9\sqrt{17}$

Explanation:

$\sqrt{(45)(9)+(27)(36)}$

When simplifying a square root, consider the factors of each of its component parts:

$\sqrt{(3^2)\times5\times(3^{2})+(3^{3})(2^2)(3^2)}$

Combine like terms:

$\sqrt{5(3^4)+(2^2)(3^5)}$

Remove the common factor, 3^4 :

$\sqrt{(3^4)\times(5+3\times(2^2))}$

Pull the $\sqrt{3^4}$ outside of the equation as 3^2 :

$(3^2)\sqrt{5+12}=9\sqrt{17}$

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Example Question #5 : Squaring / Square Roots / Radicals

Which of the following is equal to the following expression?

$\sqrt{(16)(8)+(32)(20)}$

Possible Answers:

$2^{3}\sqrt{6}$

$2^{3}\sqrt{10}$

$2^4\sqrt{3}$

$2^{7}\sqrt{5}$

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

Correct answer:

$2^4\sqrt{3}$

Explanation:

$\sqrt{(16)(8)+(32)(20)}$

First, break down the components of the square root:

$\sqrt{(2^{4})(2^{3})+(2^{5})(2^{2})\times 5}$

Combine like terms. Remember, when multiplying exponents, add them together:

$\sqrt{(2^{7})+(2^{7})\times5}$

Factor out the common factor of 2^7 :

$\sqrt{(2^{7})(1+5)}$

$\sqrt{(2^7)\times6}$

Factor the :

$\sqrt{(2^7)\times2\times3}$

Combine the factored with the 2^7 :

$\sqrt{(2^{8})\times3}$

Now, you can pull $\sqrt{2^8}$ out from underneath the square root sign as 2^4 :

$2^4\sqrt{3}$

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Example Question #6 : Squaring / Square Roots / Radicals

Which of the following expressions is equal to the following expression?

$\sqrt{(27)(45)(125)}$

Possible Answers:

$125\sqrt{27}$

$205\sqrt{3}$

$225\sqrt{3}$

$75\sqrt{20}$

$135\sqrt{5}$

Correct answer:

$225\sqrt{3}$

Explanation:

$\sqrt{(27)(45)(125)}$

First, break down the component parts of the square root:

$\sqrt{(3^{3})(5\times 3^{2})(5^{3})}$

Combine like terms in a way that will let you pull some of them out from underneath the square root symbol:

$\sqrt{(5^{4})(3^4)(3)}$

Pull out the terms with even exponents and simplify:

$(5^{2})(3^{2})\sqrt{3}=225\sqrt{3}$

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Example Question #4 : How To Factor A Common Factor Out Of Squares

What is,

$\frac{12}{\sqrt{18}}$ ?

Possible Answers:

$3*\sqrt{2}$

$2*\sqrt{3}$

$4*\sqrt{3}$

$2*\sqrt{2}$

$3*\sqrt{3}$

Correct answer:

$2*\sqrt{2}$

Explanation:

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

$\frac{12}{\sqrt{18}}*\frac{\sqrt{18}}{\sqrt{18}}=$

$\frac{12*\sqrt{18}}{18}=$

Factor out 6,

$\frac{2*\sqrt{18}}{3}=$

Extract perfect square 9 from the square root of 18.

$\sqrt{18}=3*\sqrt{2}$

$\frac{2*3*\sqrt{2}}{3}=$

$2*\sqrt{2}$

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ACT Math : Factoring Squares

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Factoring Squares

Example Question #2 : How To Factor A Common Factor Out Of Squares

Example Question #2 : How To Factor A Common Factor Out Of Squares

Example Question #4 : Squaring / Square Roots / Radicals

Example Question #5 : Squaring / Square Roots / Radicals

Example Question #6 : Squaring / Square Roots / Radicals

Example Question #4 : How To Factor A Common Factor Out Of Squares

All ACT Math Resources

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