Evaluate this complex number expression:
| 8 | |
| 3 + 4i |
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:
| 8 | |
| 3 + 4i |
| = |
| 24 - 32i |
| 25 |
This fraction cannot be reduced down anymore, so we have our answer
| 8 | |
| 3 + 4i |
| = |
| 24 - 32i |
| 25 |
Final Answer
| 8 | |
| 3 + 4i |
| = |
| 24 - 32i |
| 25 |
Common Core State Standards In This Lesson
HSN.CN.A.2, HSN.CN.A.3
What is the Answer?
| 8 | |
| 3 + 4i |
| = |
| 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.
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.
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.
What 8 concepts are covered in the Complex Number Operations Calculator?
- absolute value
- A positive number representing the distance from 0 on a number line
- addition
- math operation involving the sum of elements
- complex number
- a number that can be written in the form a + b or a - bi
- complex number operations
- conjugate
- A term formed by changing the sign between two terms in a binomial.
- division
- separate a number into parts
- multiplication
- math operation involving the product of elements
- subtraction
- math operation involving the difference of elements
