%2F(8%2B2i).jpeg "(3-5i)/(8+2i)")
Simplifying Complex Fractions: (3-5i)/(8+2i)
This article will guide you through the process of simplifying the complex fraction (3-5i)/(8+2i). The key to simplifying complex fractions is to eliminate the imaginary term from the denominator.
Multiplying by the Conjugate
The conjugate of a complex number is found by changing the sign of the imaginary part. In our case, the conjugate of (8+2i) is (8-2i).
To eliminate the imaginary term in the denominator, we multiply both the numerator and denominator by the conjugate:
(3-5i)/(8+2i) * (8-2i)/(8-2i)
Expanding the Multiplication
Now, we need to expand the multiplication in both the numerator and denominator. Remember that i² = -1:
Numerator: (3-5i)(8-2i) = 24 - 6i - 40i + 10i² = 24 - 46i - 10 = 14 - 46i
Denominator: (8+2i)(8-2i) = 64 - 16i + 16i - 4i² = 64 + 4 = 68
Simplifying the Result
Now our fraction looks like this: (14-46i)/68
We can further simplify this by dividing both the numerator and denominator by their greatest common factor, which is 2:
(14-46i)/68 = (7-23i)/34
Final Result
Therefore, the simplified form of the complex fraction (3-5i)/(8+2i) is (7-23i)/34.