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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