Evaluate this complex number expression:

Find the conjugate

If the denominator is c + di:
The conjugate is c - di.

Multiply by the conjugate

(8)(3 - 4i)
(3 + 4i)(3 - 4i)

Expand the denominator

(3 + 4i)(3 - 4i)

Define the FOIL Formula:

(a * c) + (b * c) + (a * d) + (b * d)

Set the FOIL values:

a = 3, b = 4, c = 3, and d = -4

Plug in values:

(3 + 4i)(3 - 4i) = (3 * 3) + (4i * 3) + (3 * -4i) + (4i * -4i)

(3 + 4i)(3 - 4i) = 9 + 12i - 12i - 16i2

Group the like terms:

(3 + 4i)(3 - 4i) = 9 + (12 - 12)i - 16i2

(3 + 4i)(3 - 4i) = 9 - 16i2

Simplify our last term:

i2 = √-1 * √-1 = -1, so our last term becomes:

(3 + 4i)(3 - 4i) = 9 - 16* (-1)

(3 + 4i)(3 - 4i) = 9 + 16

Group the 2 constants

(3 + 4i)(3 - 4i) = (9 + 16)

Expand the numerator

(8)(3 - 4i)

Define the FOIL Formula:

(a * c) + (b * c) + (a * d) + (b * d)

Set the FOIL values:

a = 8, b = 0, c = 3, and d = -4

Plug in values:

(8)(3 - 4i) = (8 * 3) + (8 * -4i)

(8)(3 - 4i) = 24 - 32i

After expanding and simplifying numerator and denominator, we are left with:

24 - 32i

25

This fraction cannot be reduced down anymore, so we have our answer

24 - 32i

25

Final Answer

24 - 32i

25

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What is the Answer?

24 - 32i

25

How does the Complex Number Operations Calculator work?

Free Complex Number Operations Calculator - Given two numbers in complex number notation, this calculator:
1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.
2) Determines the Square Root of a complex number denoted as √a + bi
3) Absolute Value of a Complex Number |a + bi|
4) Conjugate of a complex number a + bi
This calculator has 4 inputs.

What 6 formulas are used for the Complex Number Operations Calculator?

a + bi + (c + di) = (a + c) + (b + d)i
a + bi - (c + di) = (a - c) + (b - d)i
(a * c) + (b * c) + (a * d) + (b * d)
The square root of a complex number a + bi, is denoted as root1 = x + yi and root2 = -x - yi
|a + bi| = sqrt(a2 + b2)
a + bi has a conjugate of a - bi and a - bi has a conjugate of a + bi.

For more math formulas, check out our Formula Dossier

What 8 concepts are covered in the Complex Number Operations Calculator?

absolute valueA positive number representing the distance from 0 on a number lineadditionmath operation involving the sum of elementscomplex numbera number that can be written in the form a + b or a - bicomplex number operationsconjugateA term formed by changing the sign between two terms in a binomial.divisionseparate a number into partsmultiplicationmath operation involving the product of elementssubtractionmath operation involving the difference of elements

Example calculations for the Complex Number Operations Calculator

Complex Number Operations Calculator Video

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