Factoring Squares

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ACT Math › Factoring Squares

Questions 1 - 10

1

Which of the following expression is equal to

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

$9\sqrt{17}$

$9\sqrt{19}$

$7\sqrt{17}$

$9\sqrt{15}$

$7\sqrt{15}$

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, :

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

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

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

2

Which of the following expression is equal to

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

$9\sqrt{17}$

$9\sqrt{19}$

$7\sqrt{17}$

$9\sqrt{15}$

$7\sqrt{15}$

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, :

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

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

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

3

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

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

$225\sqrt{3}$

$125\sqrt{27}$

$75\sqrt{20}$

$135\sqrt{5}$

$205\sqrt{3}$

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:

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

4

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

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

$225\sqrt{3}$

$125\sqrt{27}$

$75\sqrt{20}$

$135\sqrt{5}$

$205\sqrt{3}$

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:

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

5

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

Explanation

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

(3x)(9)=272

= (3x)(32)=(33)2

=(3x+2)=36

Therefore:

x+2=6

x=4

6

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

Explanation

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

(3x)(9)=272

= (3x)(32)=(33)2

=(3x+2)=36

Therefore:

x+2=6

x=4

7

Which of the following is equal to the following expression?

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

$2^4\sqrt{3}$

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

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

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

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

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 :

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

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

Factor the :

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

Combine the factored with the :

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

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

$2^4\sqrt{3}$

8

Which of the following is equal to the following expression?

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

$2^4\sqrt{3}$

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

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

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

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

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 :

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

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

Factor the :

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

Combine the factored with the :

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

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

$2^4\sqrt{3}$

9

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

$x^{2}+ 9$

$x^{2}+ 6$

$2x^{2}+ 9$

$x^{2}+ 3$

$2x^{2}+ 3$

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.

10

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

$x^{2}+ 9$

$x^{2}+ 6$

$2x^{2}+ 9$

$x^{2}+ 3$

$2x^{2}+ 3$

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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