Question 1

Which of the following is the standard form of the complex number resulting from the operation $$ (5 - 2i) + (3 + 3i) $$?

Accepted Answer

$$ 8 + i $$

Question 2

What is the result of $$ (2 + 6i) - (12 - i)  in $$standard form?

Accepted Answer

$$ -10 + 7i $$

Question 3

Calculate the product $$ 7i(2 - 9i) $$ and write your answer in standard form.

Accepted Answer

$$ 63 + 14i $$

Question 4

Find the product $$ (5 + 4i)(6 - 7i) $$ and express your answer in standard form.

Accepted Answer

$$ 58 - 11i $$

Question 5

What is the complex conjugate of $$ a + bi $$?

Accepted Answer

$$ a - bi $$

Question 6

What is the result of multiplying a complex number by its conjugate, $$ (a + bi)(a - bi) $$?

Accepted Answer

$$ a^2$ + b^2$ $$

Question 7

Divide $$ \dfrac{5 + 4i}{4 - i} $$ and write your answer in standard form.

Accepted Answer

$$ \dfrac{16 + 21i}{17} $$

Question 8

What is the principal square root of $$ -27 $$?

Accepted Answer

$$ 3i\sqrt{3} $$

Question 9

Simplify $$ \sqrt{-27} + \sqrt{-48} $$ and write your answer in terms of $$ i . $$

Accepted Answer

$$ 7i\sqrt{3} $$

Question 10

Evaluate $$ (-2 - \sqrt{3})^2$ $$ and write your answer in standard form.

Accepted Answer

$$ 1 + 4i\sqrt{3} $$

Question 11

Which of the following statements about complex numbers is TRUE?

Accepted Answer

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

Question 12

If $$ i  is $$the imaginary unit, what is $$ i^2$ $$?

Accepted Answer

$$ -1 $$

Question 13

Which of the following is NOT a complex number?

Accepted Answer

$$ 	ext{None of the above} $$

Question 14

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

Accepted Answer

$$ 5 $$

Question 15

Which of the following best describes the set of all complex numbers?

Accepted Answer

All numbers of the form $$ a + bi $$ where $$ a $$ and $$ b $$ are real numbers.

Question 16

A circuit engineer needs to add the signals $$ (2 + 5i) $$ and $$ (3 - 2i) . $$What is the resulting signal?

Accepted Answer

$$ 5 + 3i $$

Question 17

Which operation is used to rationalize the denominator when dividing complex numbers?

Accepted Answer

Multiplying numerator and denominator by the conjugate of the denominator.

Question 18

If $$ z = 3 + 4i $$, what is $$ z \cdot \overline{z} $$?

Accepted Answer

$$ 25 $$

Question 19

Which of the following is the correct simplification for $$ \sqrt{-16} $$?

Accepted Answer

$$ 4i $$

Complex Numbers: Operations and Properties

Study Guide - Practice Questions