how do I solve (8+i)/(1+2i) ?

Question

I understand that I need to multiply the top and bottom by something, but I can't seem to figure out how to get the answer that was named as 2i-3

Answer 1

You are correct, you do need to multiply the top and bottom by something, and this "something" is called the conjugate, does that ring a bell? If not, remember, the conjugate that needs to be multiplied in this case is the denominator. The conjugate is the opposite in signs of the denominator, so: 

(8+i)/(1+2i)

Multiply the top and bottom by the denominator conjugate (1-2i).

(8+i)/(1+2i)*(1-2i)/(1-2i)

Now, multiply out the denominator by FOILing. This leaves: 

(8+i)(1-2i)/5

Then you can multiply the top by FOILing.

8−16i+i−2i^2/5

To get rid of the fraction, multiply.

(8−15i−2i^2)/5

Then you must realize that i raised to the 2nd power is equal to -1. This means that you can simplify further.

8−15i+2/5

Remove the fraction by multiplication. 

1/5(8−15i+2)

Add 2 and 8 to get 10.

1/5(10-15i)

 Divide out the fraction

10/5-15i/5

Cancel the common factors (2 in 10/5 and 3 in 15i/5) which leaves: 

2-3i 

The problem is finished!