Squaring / Square Roots / Radicals - ACT Math

Card 0 of 414

Question

The expression $\frac{x^2-25}{x^2+11x+30}$ is equivalent to:

Answer

First, we need to factor the numerator and denominator separately and cancel out similar terms. We will start with the numerator because it can be factored easily as the difference of two squares.

Now factor the quadratic in the denominator.

Substitute these factorizations back into the original expression.

$\frac{(x+5)(x-5)}{(x+5)(x+6)}$

The terms cancel out, leaving us with the following answer:

$\frac{x-5}{x+6}$

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Question

Expand:

Answer

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

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Question

$\small (3a-2b)^{2}$ can be rewritten as:

Answer

Use the formula for solving the square of a difference, $\small (x-y)^{2}=x^{2}-2xy+y^{2}$ . In this case, $\small \small \small (3a-2b)^{2}=9a^{2}-12ab+4b^{2}$

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Question

Evaluate the following expression:

$2^{5}+(5*2)^{2}$

Answer

2 raised to the power of 5 is the same as multiplying 2 by itself 5 times so:

25 = 2x2x2x2x2 = 32

Then, 5x2 must first be multiplied before taking the exponent, yielding 102 = 100.

100 + 32 = 132

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Question

Expand:

Answer

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

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Question

Which of the following is the square of ?

Answer

Use the square of a sum pattern, substituting for and for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(4x + 7y)^{2}= (4x)^{2}+ 2 (4x) (7y) + (7y)^{2}$

$= 16x^{2}+ 56xy + 49y^{2}$

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Question

Which of the following is the square of $5 \sqrt{x} +3 \sqrt{y}$ ?

You may assume both and are positive.

Answer

Use the square of a sum pattern, substituting $5 \sqrt{x}$ for and $3 \sqrt{y}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(5 \sqrt{x}+3 \sqrt{y})^{2}= (5 \sqrt{x})^{2}+ 2 (5 \sqrt{x})\left (3 \sqrt{y} \right ) + \left (3 \sqrt{y} \right )^{2}$

$(5 \sqrt{x}+3 \sqrt{y})^{2}= 25x+ 30 \sqrt{xy} + 9y$

or

$25x + 9y+ 30 \sqrt{xy}$

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Question

Which of the following is the square of $x^{2}+ x + 7$ ?

Answer

Multiply vertically as follows:

$\begin{matrix} x^{2}+ ; ; ; x +; ; ; 7\\underline{x^{2}+; ; ; x +; ; ; 7} \end{matrix}$

$7 x^{2}+7 x +49$

$x^{3}+ x^{2} + 7x$

$\underline{x^{4}+ x^{3}+7 x^{2} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; }$

$x^{4}+2x^{3}+15x^{2}+14x+49$

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Question

Which of the following is the square of $y + \sqrt{17}$ ?

Answer

Use the square of a sum pattern, substituting for and $\sqrt{17}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + (\sqrt{17})^{2}$

$(y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + 17$

This is not equivalent to any of the given choices.

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Question

Which of the following is the square of $\frac{1}{4}x + \frac{1}{7}$ ?

Answer

Use the square of a sum pattern, substituting $5 \sqrt{x}$ for and $3 \sqrt{y}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$\left (\frac{1}{4}x + \frac{1}{7} \right )^{2}= \left (\frac{1}{4}x \right )^{2}+ 2 \left (\frac{1}{4}x \right )\left (\frac{1}{7} \right ) + \left (\frac{1}{7} \right )^{2}$

$= \frac{1}{16}x ^{2}+ \frac{1}{14}x +\frac{1}{49}$

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Question

Which of the following is the square of ?

Answer

Use the square of a sum pattern, substituting for and for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$\left (3.7x + 1.4 y \right )^{2}= (3.7x)^{2}+ 2 (3.7x)(1.4 y) + (1.4 y) ^{2}$

$= 13.69x^{2}+ 10.36x y +1.96 y ^{2}$

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Question

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

Answer

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

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Question

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

Answer

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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Question

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

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

Answer

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

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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Question

Which of the following is equal to the following expression?

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

Answer

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

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 :

$2^4\sqrt{3}$

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Question

Which of the following expression is equal to

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

Answer

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

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 :

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

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Question

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

Answer

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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Question

What is,

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

Answer

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.

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

$\frac{23\sqrt{2}}{3}=$

$2*\sqrt{2}$

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Question

Suppose and

Evaluate the following expression:

$3a^3+\frac{1}{2}(2b^2)$

Answer

Substituting for and , we have

$3(-2)^{3}+\frac{1}{2}(2(4)^{2})$

This simplifies to

which equals

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Question

What is the sum of and given

and

?

Answer

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation , is the real component and is the imaginary component (designated by ).

In equation , is the real component and is the imaginary component.

When added,

$a + b = \left ( 5+3i\right ) + \left ( 2 + i\right ) = \left ( 5 + 2\right ) + i\left ( 3 + 1\right ) = 7 + 4i$

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