(5+4i)/(1-3i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (5 + 4i) / (1 - 3i)

In the realm of complex numbers, division takes a slightly different approach than with real numbers. To divide complex numbers, we employ a clever technique using the conjugate of the denominator.

Here's how we simplify (5 + 4i) / (1 - 3i):

1. Identify the conjugate: The conjugate of (1 - 3i) is (1 + 3i). We obtain the conjugate by simply changing the sign of the imaginary part.

2. Multiply both numerator and denominator by the conjugate:

   (5 + 4i) / (1 - 3i) * (1 + 3i) / (1 + 3i)

3. Expand using the distributive property (or FOIL method):

   (5 + 15i + 4i + 12i²) / (1 + 3i - 3i - 9i²)

4. Simplify using the fact that i² = -1:

   (5 + 19i - 12) / (1 + 9)

5. Combine real and imaginary terms:

   (-7 + 19i) / 10

6. Express the result in standard form:

   -7/10 + 19/10i

Therefore, the simplified form of (5 + 4i) / (1 - 3i) is -7/10 + 19/10i.