Simplify three minus six 𝑖 divided by one minus five 𝑖.
When working with imaginary numbers and complex numbers, you cannot have an 𝑖 on a denominator. So here, to get rid of the 𝑖 on the denominator, we have to multiply by the complex conjugate. So the complex conjugate is where you keep the first number the same and you change the sign of the second number. So instead of one minus five 𝑖, the complex conjugate is one plus five 𝑖.
And whatever we do to the denominator, we have to also do to the numerator. So now we need to FOIL. We need to use the distributive property. Three times one is three. Three times five 𝑖 is 15𝑖. Negative six 𝑖 times one is negative six 𝑖. And negative six 𝑖 times five 𝑖 is equal to negative 30𝑖 squared.
Now for the denominator, one times one is one, one times five 𝑖 is five 𝑖, negative five 𝑖 times one is negative five 𝑖, and negative five 𝑖 times positive five 𝑖 is negative 25𝑖 squared. However, 𝑖 squared is equal to negative one.
The reason why is because 𝑖 squared is equal to 𝑖 times 𝑖. And 𝑖 is equal to square root negative one. So 𝑖 times 𝑖 is the same thing as square root negative one times square root negative one, which would be negative one.
So again 𝑖 squared is equal to negative one. So let’s replace these with negative one. And multiplying negative 30 times negative one is positive 30 and negative 25 times negative one is positive 25. So now we can combine the 𝑖s and we can combine the constant numbers. On the numerator three plus 30 is 33, and 15𝑖 minus six 𝑖 is nine 𝑖.
On the denominator, one plus 25 is 26 and then five 𝑖 minus five 𝑖 is zero 𝑖, which we do not have to write plus zero 𝑖. We can leave it out. And splitting this into two fractions would be thirty-three twenty-sixths plus nine 𝑖 twenty-sixths.