How do you solve for an operation with complex numbers. Example: 4i over -3+6i

You can clear the comnplex number from the denominator by multiplying both top and bottom by the complex conjugate of the number on the bottom.  Using your example we have

$$\frac{4i}{-3+6i}$$

 

The complex conjugate of -3 + 6i is -3 - 6i, so

$$\frac{4i}{-3+6i}=\frac{4i(-3-6i)}{(-3+6i)(-3-6i)}$$

 

Now the denominator can be expressed as the difference of two squares:  (-3)² - (6i)² = 9 - 36i² = 9 + 36 = 45

and the numerator becomes -3*4i - 6i*4i = -12i -24i² = -12i +24 (or 24 - 12i)

 

Hence:

$$\frac{4i}{-3+6i}=\frac{24-12i}{45}$$

 

or $$\frac{4i}{-3+6i}=\frac{8-4i}{15} = \frac{8}{15}-\frac{4i}{15}$$

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