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ACT Math : Exponents

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Example Question #1 : How To Factor A Common Factor Out Of Squares

Which of the following expression is equal to

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

Possible Answers:

$7\sqrt{15}$

$9\sqrt{17}$

$9\sqrt{15}$

$9\sqrt{19}$

$7\sqrt{17}$

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 : How To Factor A Common Factor Out Of Squares

Which of the following is equal to the following expression?

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

Possible Answers:

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

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

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

$2^4\sqrt{3}$

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

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 : How To Factor A Common Factor Out Of Squares

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

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

Possible Answers:

$225\sqrt{3}$

$125\sqrt{27}$

$135\sqrt{5}$

$205\sqrt{3}$

$75\sqrt{20}$

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 #2321 : Act Math

What is,

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

Possible Answers:

$3*\sqrt{3}$

$4*\sqrt{3}$

$3*\sqrt{2}$

$2*\sqrt{3}$

$2*\sqrt{2}$

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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Example Question #1 : Complex Numbers

Subtract from , given:

a = 3 + i

b = 4 - 2i

Possible Answers:

-1 - 3i

1 - 3i

3 + i

-1 + 3i

1+ 3i

Correct answer:

1 - 3i

Explanation:

A complex number is a combination of a real and imaginary number. To subtract complex numbers, subtract 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. Solving for b - a ,

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

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Example Question #1 : Complex Numbers

Simplify the exponent,

$(3^{6})^{2}$ .

Possible Answers:

$3^{3}$

$3^{12}$

$3^{8}$

$3^{4}$

Correct answer:

$3^{12}$

Explanation:

When you have an exponent on the outside of parentheses while another is on the inside of the parentheses, such as in $(3^{6})^{2}$ , multiply the exponents together to get the answer: $3^{12}$ .

This is different than when you have two numbers with the same base multiplied together, such as in $x^{2} \cdot x^{3} = x^{5}$ . In that case, you add the exponents together.

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Example Question #3 : Complex Numbers

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

Simplify:

(2-2i) - (4-i)

Possible Answers:

6-i

-2-i

6-3i

$\textup{None of these}$

-2-3i

Correct answer:

-2-i

Explanation:

Solving this equation is very similar to solving a linear binomial like (a+bx) . To solve, just combine like terms, being careful to watch for double negatives.

(2-2i) - (4-i)

(2-4) + (-2i-(-i))

-2 -(-2i+i)

-2 -i

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Example Question #4 : Complex Numbers

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?

Possible Answers:

-(4-i) = (4-i) = 0

2i - (2-2i) = -2 +4i

-3i - (7-7i) = -7 +4i

(7-i) -(2+i) = 5-2i

7i- (3-i) = 3-8i

Correct answer:

7i- (3-i) = 3-8i

Explanation:

A problem like this can be solved similarly to a linear binomial like (a +bx) /

7i - (3-i) = 3-8i

7i + (-3 + i) = 3-8i

$-3 +8i \neq 3+8i$

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Example Question #5 : Complex Numbers

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

Which of the following equations simplifies into 4+2i ?

Possible Answers:

(0 + 7i) - (-4 - 5i)

3 - (7 + 2i)

(8-5i) + (-4 +7i)

(10 -i) - (6 - i)

0+ 6i + (-4 - 4i)

Correct answer:

(8-5i) + (-4 +7i)

Explanation:

This equation can be solved very similarly to a binomial like a+bx .

(8-5i) + (-4 +7i)

(8-4) + (-5i + 7i)

4 + 2i

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Example Question #6 : Complex Numbers

Suppose a=-2 and b=4

Evaluate the following expression:

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

Possible Answers:

Correct answer:

Explanation:

Substituting for and , we have

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

This simplifies to

-24+16

which equals

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ACT Math : Exponents

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Factor A Common Factor Out Of Squares

Example Question #5 : How To Factor A Common Factor Out Of Squares

Example Question #6 : How To Factor A Common Factor Out Of Squares

Example Question #2321 : Act Math

Example Question #1 : Complex Numbers

Example Question #1 : Complex Numbers

Example Question #3 : Complex Numbers

Example Question #4 : Complex Numbers

Example Question #5 : Complex Numbers

Example Question #6 : Complex Numbers

All ACT Math Resources

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