# How to Multiply and Divide Complex Numbers? (+FREE Worksheet!)

Learn how to multiply and divide complex numbers into a few simple steps using the following step-by-step guide.

## Step-by-step Guide to Multiplying and Dividing Complex Numbers

• Multiplying complex numbers: $$\color{blue}{(a+bi)×(c+di)=(ac-bd)+(ad+bc)i}$$
• Dividing complex numbers: $$\color{blue}{\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2}i }$$
• Imaginary number rule: $$\color{blue}{i^2=-1}$$

### Multiplying and Dividing Complex Numbers – Example 1:

Solve: $$\frac{4-2i}{2+i}=$$

Solution:

Use the rule for dividing complex numbers:
$$\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2 } i→$$
$$\frac{4-2i}{2+i}×\frac{2-i}{2-i}=\frac{(4)×(2)+(-2) × (1)}{2^2+ (1)^2 }+\frac{(-2)×(2)-(4) × (1)}{2^2+(1)^2 } i=\frac {8-2} {5}+ \frac{-4-4} {5}i=\frac{6}{5}-\frac{8}{5} i$$

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### Multiplying and Dividing Complex Numbers – Example 2:

Solve: $$(2-3i)(4-3i)$$

Solution:

Use the rule: $$(a+bi) × (c+di)=(ac-bd)+(ad+bc)i$$
$$((2) × (4)-(-3)(-3))+((2) × (-3)+(-3)(4))i= (8-9)+(-6-12)i=-1-18i$$

### Multiplying and Dividing Complex Numbers – Example 3:

Solve: $$(2-8i)(3-5i)$$

Solution:

Use the rule: $$(a+bi) × (c+di)=(ac-bd)+(ad+bc)i$$
$$((2) × (3)-(-8) × (-5))+((2) × (-5)+(-8) × (3))i=(6-40)+(-10-24)i=-34-34i$$

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### Multiplying and Dividing Complex Numbers – Example 4:

Solve: $$\frac{2-3i}{2+i}=$$

Solution:

Use the rule for dividing complex numbers:
$$\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2 } i→$$
$$\frac{2-3i}{2+i}×\frac{2-i}{2-i}=\frac{(2)×(2)+(-3) × (1)}{2^2+ (-1)^2 }+\frac{(-3)×(2)+(-2) × (1)}{2^2+(-1)^2 } i=\frac{4-3}{5}-\frac {-6-2}{5}i=\frac{1}{5}-\frac{8}{5} i$$

## Exercises for Multiplying and Dividing Complex Numbers

### Simplify.

• $$\color{blue}{(4i)(– i)(2 – 5i)}$$
• $$\color{blue}{(3 – 7i)(4 – 5i)}$$
• $$\color{blue}{(–5 + 9i)(3 + 5i)}$$
• $$\color{blue}{(7 + 3i)(7+ 8i)}$$
• $$\color{blue}{(5 + 4i)^2}$$
• $$\color{blue}{2(3i) – (5i)(– 8 + 5i)}$$

• $$\color{blue}{8 – 20i }$$
• $$\color{blue}{–23 – 43i}$$
• $$\color{blue}{–60 + 2i}$$
• $$\color{blue}{25 + 77i}$$
• $$\color{blue}{9 + 40i}$$
• $$\color{blue}{25 + 46i}$$

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