Question 1

Which of the following is the correct definition of the imaginary unit $$i$$?

Accepted Answer

$$i = \sqrt{-1}$$

Question 2

What is the standard form of a complex number?

Accepted Answer

$$a + bi$$

Question 3

If $$z_1 = 3 + 4i$ and z_2$ = 2 - 5i$$, what is $$z_1$$ + $$z_2$$?

Accepted Answer

$$1 - i$$

Question 4

Which of the following is the complex conjugate of $$7 - 2i$$?

Accepted Answer

$$7 + 2i$$

Question 5

Simplify the product $$(2 + 3i)(4 - i)$$ and write the result in standard form.

Accepted Answer

$$8 - 2i + 12i - 3i^2$ = 11 + 10i$$

Question 6

A medical imaging device uses the complex number $$z = 5 - 12i to $$represent a signal. What is the modulus (absolute value) of $$z$$?

Accepted Answer

$$\sqrt{5^2$ + 12^2$} = 13$$

Question 7

Simplify $$i^{23}.$$

Accepted Answer

$$i$$

Question 8

Which of the following is the result of dividing $$3 + 2i by$$ $$1 - i$$ and expressing the answer in standard form?

Accepted Answer

$$\frac{3 + 2i}{1 - i} = \frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} = \frac{3 + 3i + 2i + 2i^2$}{1^2$ - i^2$} = \frac{3 + 5i - 2}{1 + 1} = \frac{1 + 5i}{2} = 0.5 + 2.5i$$

Question 9

A chemical reaction is modeled by the complex number $$z = -6 + 8i. $$What is the complex conjugate of $$z$$?

Accepted Answer

$$-6 - 8i$$

Question 10

Simplify $$\sqrt{-49}$$ and write the result in standard form.

Accepted Answer

$$7i$$

Question 11

Which of the following is true about the product of a complex number and its conjugate, $$z \cdot \overline{z}$$, where $$z = a + bi$$?

Accepted Answer

$$z \cdot \overline{z} = a^2$ + b^2$

Question 12

A signal in an electrical circuit is represented by $$z = 2 - 3i. $$What is $$z - (5 + 4i)$$?

Accepted Answer

$$(2 - 5) + (-3 - 4)i = -3 - 7i$$

Question 13

Simplify the expression $$\sqrt{-16} + \sqrt{-9}.$$

Accepted Answer

$$4i + 3i = 7i$$

Question 14

Which of the following is the correct result for $$(1 + 2i)(3 - 4i)$$?

Accepted Answer

$$3 - 4i + 6i - 8i^2$ = 3 + 2i + 8 = 11 + 2i$$

Question 15

If $$i^2 = -1$, what is i$$^{72}$$?

Accepted Answer

$$1$$

Question 16

A physicist models a wave using $$z = 6i. $$What is the result of $$z^2$$?

Accepted Answer

$$(6i)^2$$ = $$36i^2 = -36$

Question 17

Simplify $$\frac{4 + 3i}{2 - i}$$ and write the result in standard form.

Accepted Answer

$$\frac{(4 + 3i)(2 + i)}{(2 - i)(2 + i)} = \frac{8 + 4i + 6i + 3i^2$}{4 + 2i - 2i - i^2$} = \frac{8 + 10i - 3}{4 + 1} = \frac{5 + 10i}{5} = 1 + 2i$$

Question 18

Which of the following is the correct simplification of $$\sqrt{-25} \cdot \sqrt{-4}$$?

Accepted Answer

$$(5i) \cdot (2i) = 10i^2$ = -10$$

Question 19

A signal in a communications system is represented by $$z = 4 + 3i. $$What is the result of $$z \cdot \overline{z}$$?

Accepted Answer

$$(4 + 3i)(4 - 3i) = 16 - 12i + 12i - 9i^2$ = 16 + 9 = 25$$

Complex Numbers in College Algebra: Definitions, Operations, and Applications

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