Evaluate this complex number expression:
| 5 - 7i | |
| 7 + 3i |
Find the conjugate
If the denominator is c + di:
The conjugate is c - di.
Multiply by the conjugate
| (5 - 7i)(7 - 3i) | |
| (7 + 3i)(7 - 3i) |
Expand the denominator
(7 + 3i)(7 - 3i)
Define the FOIL Formula:
(a * c) + (b * c) + (a * d) + (b * d)
Set the FOIL values:
a = 7, b = 3, c = 7, and d = -3
Plug in values:
(7 + 3i)(7 - 3i) = (7 * 7) + (3i * 7) + (7 * -3i) + (3i * -3i)
(7 + 3i)(7 - 3i) = 49 + 21i - 21i - 9i2
Group the like terms:
(7 + 3i)(7 - 3i) = 49 + (21 - 21)i - 9i2
(7 + 3i)(7 - 3i) = 49 - 9i2
Simplify our last term:
i2 = √-1 * √-1 = -1, so our last term becomes:
(7 + 3i)(7 - 3i) = 49 - 9* (-1)
(7 + 3i)(7 - 3i) = 49 + 9
Group the 2 constants
(7 + 3i)(7 - 3i) = (49 + 9)
Expand the numerator
(5 - 7i)(7 - 3i)
Define the FOIL Formula:
(a * c) + (b * c) + (a * d) + (b * d)
Set the FOIL values:
a = 5, b = -7, c = 7, and d = -3
Plug in values:
(5 - 7i)(7 - 3i) = (5 * 7) + (-7i * 7) + (5 * -3i) + (-7i * -3i)
(5 - 7i)(7 - 3i) = 35 - 49i - 15i + 21i2
Group the like terms:
(5 - 7i)(7 - 3i) = 35 + (-49 - 15)i + 21i2
(5 - 7i)(7 - 3i) = 35 - 64i + 21i2
Simplify our last term:
i2 = √-1 * √-1 = -1, so our last term becomes:
(5 - 7i)(7 - 3i) = 35 - 64i + 21* (-1)
(5 - 7i)(7 - 3i) = 35 - 64i - 21
Group the 2 constants
(5 - 7i)(7 - 3i) = (35 - 21) - 64i
After expanding and simplifying numerator and denominator, we are left with:
| 5 - 7i | |
| 7 + 3i |
| = |
| 14 - 64i |
| 58 |
Our fraction is not fully reduced
The Greatest Common Factor (GCF) of 14, -64, and 58 is 2
Reducing our fraction by the GCF, we get our answer:
| 5 - 7i |
| 7 + 3i |
| = |
| 7 - 32i |
| 29 |
| 5 - 7i | |
| 7 + 3i |
| = |
| 7 - 32i |
| 29 |
Final Answer
| 5 - 7i | |
| 7 + 3i |
| = |
| 7 - 32i |
| 29 |
Common Core State Standards In This Lesson
What is the Answer?
| 5 - 7i | |
| 7 + 3i |
| = |
| 7 - 32i |
| 29 |
How does the Complex Number Operations Calculator work?
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 * 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
