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ACT Math : Algebra

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Example Questions

Example Question #5 : Complex Numbers

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

Simplify: (5 +7i) + (17-4i)

Possible Answers:

22 +11i

-12 +3i

-12-3i

22-3i

22+3i

Correct answer:

22+3i

Explanation:

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

(5 +7i) + (17-4i) = (5+17) + (7i - 4i) = 22 + 3i

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Example Question #6 : Complex Numbers

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

Can you add the following two numbers: $3+7i \textup{ and } 8$ ? If so, what is their sum?

Possible Answers:

$\textup{Yes, } 3 + 7i + 8$

$\textup{Yes, } 11 + 15i$

$\textup{Yes, }3 + 15i$

$\textup{Yes, } 11+ 7i$

$\textup{No, the two numbers cannot be added because one is complex and one is not.}$

Correct answer:

$\textup{Yes, } 11+ 7i$

Explanation:

Complex numbers take the form a + bi, where a is the real term in the complex number and bi is the nonreal (imaginary) term in the complex number. Taking this, we can see that for the real number 8, we can rewrite the number as 8 + 0i , where represents the (zero-sum) non-real portion of the complex number.

Thus, any real number can be added to any complex number simply by considering the nonreal portion of the number to be .

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Example Question #1 : How To Add 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 is incorrect?

Possible Answers:

(2-i) + (-2+i) = 0

7i+ (3-2i) = 3+5i

(3+7i) - (2+2i) = 1 +9i

2+ (2+5i) = 3i + (4+2i)

(3+3i) - (3-3i) = 0+ 6i

Correct answer:

(3+7i) - (2+2i) = 1 +9i

Explanation:

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.

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

(3+7i) - (2+2i) = 1 +9i

(3-2) + (7i-2i) = 1+9i

$1+5i \neq 1+9i$

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Example Question #1 : How To Divide Complex Numbers

Simplify: $\frac{9+4i}{4i}$

Possible Answers:

$\frac{ 9 }{4 } + i$

$-\frac{ 9 }{4 } + i$

$\frac{ 9 }{4 } - i$

$1- \frac{ 9 }{4 } i$

$1+ \frac{ 9 }{4 } i$

Correct answer:

$1- \frac{ 9 }{4 } i$

Explanation:

Multiply both numberator and denominator by :

$\frac{9+4i}{4i}$

$= \frac{\left (9+4i \right )\cdot i}{4i \cdot i}$

$= \frac{9i + 4i^{2}}{4 i^{2}}$

$= \frac{9i + 4 (-1)}{4 (-1)}$

$= \frac{-4+9i }{-4 }$

$= \frac{-4 }{-4 }+ \frac{ 9i }{-4 }$

$= 1- \frac{ 9 }{4 } i$

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Example Question #1 : How To Divide Complex Numbers

Evaluate: $100 \div 5i \div 5i$

Possible Answers:

100i

-4i

Correct answer:

Explanation:

First, divide 100 by as follows:

$\frac{100}{5i} = \frac{100 \cdot i}{5i \cdot i} = \frac{100 i}{5(-1)} = \frac{100 i}{-5}= -20i$

Now dvide this result by :

$\frac{-20i}{5i} = \frac{-20}{5} = -4$

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Example Question #3 : How To Divide Complex Numbers

Evaluate: $100 \div (1+i) \div (1-i)$

Possible Answers:

-25 +25i

25+ 25i

25-25i

Correct answer:

Explanation:

First, divide 100 by 1+i as follows:

$\frac{100}{1+i}$

$=\frac{100(1-i)}{\left (1+i \right )(1-i)}$

$= \frac{100-100i}{1^{2}+1^{2}}$

$= \frac{100-100i}{2}$

= 50-50i

Now, divide this by 1-i :

$\frac{ 50-50i}{1-i} = \frac{ 50 (1-i)}{1-i} = 50$

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Example Question #2 : How To Divide Complex Numbers

Evaluate: $100i \div (4i )^{2}$

Possible Answers:

$-\frac{25}{2}$

$\frac{25}{4}i$

$\frac{25}{2}i$

$- \frac{25}{4}$

$- \frac{25}{4}i$

Correct answer:

$- \frac{25}{4}i$

Explanation:

First, evaluate $(4i)^{2}$ :

$\left (4i \right )^{2} = 16 i^{2} = 16 (-1)= -16$

Now divide this into 100 i :

$\frac{100 i}{-16} =- \frac{25}{4}i$

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Example Question #5 : How To Divide Complex Numbers

Evaluate: $i \div \left ( 1+ i\right )^{2}$

Possible Answers:

$\frac{1}{2} i$

$\frac{1}{2}$

$-\frac{1}{2} i$

Correct answer:

$\frac{1}{2}$

Explanation:

First, evaluate $\left ( 1+ i\right )^{2}$ using the square pattern:

$\left ( 1+ i\right )^{2}$

$=1^{2}+ 2 \cdot 1 \cdot i+ i^{2}$

=1+2i+ (-1)

=2i

Divide this into :

$\frac{i}{2i} = \frac{1}{2}$

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Example Question #151 : Exponents

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: $\frac{6+12i}{4}$

Possible Answers:

$\frac{3}{2} + 12i$

$\frac{3}{2} + \frac{12i}{4i}$

6 + 3i

$\frac{3}{2} + 3i$

$\textup{None of these answers are correct}$

Correct answer:

$\frac{3}{2} + 3i$

Explanation:

This problem can be solved very similarly to a binomial such as a+bx . In this case, both the real and nonreal terms in the complex number are eligible to be divided by the real divisor.

$\frac{6+12i}{4} = (\frac{6}{4} + \frac{12i}{4})$

$\frac{12i}{4} = \frac{12}{4} \cdot i = 3 \cdot i = 3i$ , so

$\frac{3}{2} + 3i$

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Example Question #2341 : Act Math

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 by using conjugates: $\frac{4+2i}{-3-2i}$

Possible Answers:

$\frac{-16-2i}{5}$

$\frac{-16-2i}{13}$

$\frac{-16+2i}{-13}$

$\frac{-8-2i}{5}$

$\frac{-8-2i}{13}$

Correct answer:

$\frac{-16-2i}{13}$

Explanation:

Solving this problem using a conjugate is just like conjugating a binomial to simplify a denominator.

$\frac{(4+2i)(-3+2i)}{(-3-2i)(-3+2i)}$ Multiply both terms by the denominator's conjugate.

$\frac{(4+2i)(-3+2i)}{(-3-2i)(-3+2i)} = \frac{(4+2i)(-3+2i)}{13}$ Simplify. Note .

Combine and simplify.

Simplify the numerator.

$\frac{-16-2i}{13}$ The prime denominator prevents further simplifying.

Thus, $\frac{4+2i}{-3-2i} = \frac{-16-2i}{13}$ .

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #5 : Complex Numbers

Example Question #6 : Complex Numbers

Example Question #1 : How To Add Complex Numbers

Example Question #1 : How To Divide Complex Numbers

Example Question #1 : How To Divide Complex Numbers

Example Question #3 : How To Divide Complex Numbers

Example Question #2 : How To Divide Complex Numbers

Example Question #5 : How To Divide Complex Numbers

Example Question #151 : Exponents

Example Question #2341 : Act Math

All ACT Math Resources

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