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# Learn How to Solve Complex Expressions Tutorial - Definition, Formula, Example

## Learn How to Solve Complex Expressions - Tutorial

##### Definition:

When a number in the form a + bi (a real number plus an imaginary number) it is called as complex number. This tutorial will help you to solve complex expressions.

##### Example1 :

Simplify Complex Numbers: (2+3i)*(4-5i)

##### Given,

Complex Number 1 ==> (2+3i) Complex Number 2 ==> (4-5i)

##### Solution:
###### Step 1:

Complex number 2 is multiplied by 2, 2 * (4-5i) = 8-10i --> 1

###### Step 2:

Complex number 2 is multiplied by 3i, 3i * (4-5i) = 12i-15i2 = 12i + 15 --> 2

###### where,
i2 = -1

By adding 1 and 2 we will get, (2+3i)*(4-5i) = 23 + 2i

##### Example2 :

Simplify Complex Numbers: (8+i)*(1-2i) / (1+3i)

##### Given

Complex Number 1 ==> (8+i) Complex Number 2 ==> (1-2i) Complex Number 3 ==> (1+3i)

##### Solution:
###### Step 1:

Multiply Complex number 1 and 2,

 (8+i)*(1-2i) = 8(1) + 8(-2i) + i(1) + i(-2i) = 8 - 16i + i - 2i2 = 8 - 15i - 2(-1) = 8 - 15i + 2 = 10 - 15i
###### Step 2:

Step 1 result divided by Complex number 3,

(10-15i) / (1+3i) = ((10-15i)*(1-3i)) / ((1+3i)*(1-3i))
= (10 -30i - 15i + 45i2) / (1-9i2)
= (10 -45i + 45(-1)) / (1-9(-1))
= (10 -45i - 45) / (1+9)
= (-35 -45i) / 10
= 5 (-7 -9i) / 10
= (-7 -9i) / 2
= -3.5 -4.5i