Question

Write 6+i / 6-i in a + bi form

Answer 1

what would happen if you multiplied the entire thing by
\[\frac{6+i}{6+i}\]

Answer 2

I would get 36-1/ 36-1? I'm not sure for the bottom half

Answer 3

(6+i)^2 is not equal to 36-1
try that again

Answer 4

anyways the bottom will become a real number
we use difference of squares to get rid of the complex number on the bottom
once you do this, you will have a complex number over a real number

so distribute the real number and you will get your answer

Answer 5

1 + 12i?

Answer 6

is that your answer? i dont actually do these problems, i usually just give steps

Answer 7

I guess it is. I'm really lost though..

Answer 8

ok just looking at the numerator
(6+i)(6+i)
this is the same as
6(6+i)+i(6+i)=
what is the answer to that?
please write down your steps

Answer 9

6(6+i)+i(6+i)= 36+6i + 6i+i^2 = 36-1+12i
you go the 12i correct just not other part
so the numerator then becomes
35+12i

Answer 10

now you already did the denominator
(6+i)(6-i)=36-1=35

Answer 11

so what you are left with is
\[\frac{35+12i}{35}\]
can you finish this now?

Answer 12

Oh that is what I got. I assumed that 35/35 was 1 but Im pretty sure I got it know thankyou so much.

Answer 13

well you need to change it to a+bi form so
\[\frac{35+12i}{35}=\frac{35}{35}+\frac{12i}{35}=1+\frac{12}{35}i\]

Answer 14

Oh I see my mistake, I am supposed to add the denomenator under 12i as well. Thankyou.

Answer 15

The main idea is multiply top and bottom by the "complex conjugate" of the denominator
if you have a+bi, the complex conjugate is a-bi

(a+bi)(a-bi)= a^2 - (bi)^2 = a^2 -i^2 b^2
i^2= -1 , so you get a^2+b^2 which is a real number for the denominator