Squaring / Square Roots / Radicals

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ACT Math › Squaring / Square Roots / Radicals

Questions 1 - 10

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

You may assume both and are positive.

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

$25x + 9y+ 15\sqrt{xy}$

$5x + 3y+ 15\sqrt{xy}$

$5x + 3y+ 30\sqrt{xy}$

Explanation

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

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

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

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

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

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

You may assume both and are positive.

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

$25x + 9y+ 15\sqrt{xy}$

$5x + 3y+ 15\sqrt{xy}$

$5x + 3y+ 30\sqrt{xy}$

Explanation

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

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

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

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

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

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

You may assume both and are positive.

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

$25x + 9y+ 15\sqrt{xy}$

$5x + 3y+ 15\sqrt{xy}$

$5x + 3y+ 30\sqrt{xy}$

Explanation

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

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

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

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

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

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 :

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

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 :

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

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 :

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

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Which of the following is incorrect?

Explanation

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Which of the following is incorrect?

Explanation

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Which of the following is incorrect?

Explanation

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

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:

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

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