Question 1

Which of the following is the correct value of $$i^{23}$$?

Accepted Answer

$$i$$

Question 2

A medical imaging device uses the complex number $$z = 3 - 4i to $$represent a signal. What is the modulus (magnitude) of this signal?

Accepted Answer

$$5$$

Question 3

Simplify the expression $$\sqrt{-50}$$ and express your answer in terms of $$i.$$

Accepted Answer

$$5i\sqrt{2}$$

Question 4

A chemical reaction is modeled by the complex number $$z = 2 + 5i. $$What is the complex conjugate of $$z$$?

Accepted Answer

$$2 - 5i$$

Question 5

Which of the following is the result of $$(3 + 2i) + (4 - 5i)$$?

Accepted Answer

$$7 - 3i$$

Question 6

A circuit analysis problem gives the product $$(2 + 3i)(2 - 3i). $$What is the result?

Accepted Answer

$$13$$

Question 7

Simplify the quotient $$\frac{4 + i}{1 - 2i}$$ and express your answer in the form $$a + bi.$$

Accepted Answer

$$-0.2 + 2.2i$$

Question 8

Which of the following statements about complex numbers is TRUE?

Accepted Answer

The product of a complex number and its conjugate is always a non-negative real number.

Question 9

A signal in an engineering system is represented by $$z = 6 - 8i. $$What is $$z \cdot \overline{z}$$?

Accepted Answer

$$100$$

Question 10

Simplify $$i^{35}.$$

Accepted Answer

$$-i$$

Question 11

A quadratic equation in physics yields the solution $$x = \frac{-6 + \sqrt{-16}}{2}. $$What is the value of $$x$$?

Accepted Answer

$$-3 + 2i$$

Question 12

Which of the following is the standard form of the product $$(1 + 2i)(3 - 4i)$$?

Accepted Answer

$$11 + 2i$$

Question 13

Simplify $$\sqrt{-27}.$$

Accepted Answer

$$3i\sqrt{3}$$

Question 14

A medical device uses the number $$z = 5 + 12i. $$What is the argument (angle in radians) of $$z$$?

Accepted Answer

$$\arctan\left(\frac{12}{5}\right)$$

Question 15

Which of the following is the result of subtracting $$(2 - 3i)$$ from $$(7 + 4i)$$?

Accepted Answer

$$5 + 7i$$

Question 16

Simplify $$\frac{3}{2i}$$ and write your answer in standard form $$a + bi.$$

Accepted Answer

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

Question 17

If $$z = 4 + 3i$$ and $$w = 1 - 2i$$, what is $$z \cdot w$$?

Accepted Answer

$$10 - 5i$$

Question 18

A quadratic equation in engineering yields the solution $$x = \frac{-2 \pm \sqrt{-20}}{4}. $$What are the solutions?

Accepted Answer

$$-0.5 \pm i\sqrt{1.25}$$

Question 19

Which of the following is the correct simplification of $$i^{2024}$$?

Accepted Answer

$$1$$

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

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