I need help with dividing complex numbers when a radical is in the numerator:
$$\frac{ 2+√3i }{ 5-4i }$$

I know that you multiply by conjugate of the denominator (5+4i), and you use the foil method above and below. However, the introdcution of radical 3i is what is confusing me when i combine like terms. I know the denominator will come out to 41.
I started adding terms like this (below) and it all went downhill from there :)
\frac{ 10+8i+5sqrt{3}i+4sqrt{3}i ^{2}}{41}

Question

I need help with dividing complex numbers when a radical is in the numerator:
$$\frac{ 2+√3i  }{ 5-4i }$$

I know that you multiply by conjugate of the denominator (5+4i), and you use the foil method above and below.  However, the introdcution of radical 3i is what is confusing me when i combine like terms.  I know the denominator will come out to 41.
I started adding terms like this (below) and it all went downhill from there :)
\frac{ 10+8i+5sqrt{3}i+4sqrt{3}i ^{2}}{41}

anonymous · Answer

(2+√3i )( 5 +4i )  
The trouble is multiplication of these two ?

anonymous · Answer

yes that is my trouble
the numerator

anonymous · Answer

Ok, so let's go as normal and see what happens....

10 + ?

anonymous · Answer

10+8i?

anonymous · Answer

Ah, OK.....no, it is sqrt 3 not 3.

-> 10 + 5sqrt 3 i + ....?

anonymous · Answer

oh i see,
ok so

anonymous · Answer

AH, I see, you FOILED the other way, sorry....

anonymous · Answer

10 + 5sqrt 3 i + 8i  +...?

anonymous · Answer

$$10+5\sqrt{3}i+8i+4\sqrt{3}i ^{2}$$

anonymous · Answer

what happened?  Im doing something wrong?

anonymous · Answer

Yes, that's right...

anonymous · Answer

I am confusing myself, lol (I usually avoid computations)

anonymous · Answer

ok so now I need to get rid of the I and the i squared, correct?

anonymous · Answer

or is there something else I need to do first?

anonymous · Answer

Just get rid of the i^2 and bracket the i terms...

anonymous · Answer

You know what I mean, right?

anonymous · Answer

In general, you want to end up with something in the form a +ib

anonymous · Answer

$$(10+5\sqrt{3}i) (8i+4\sqrt{3}(-1))$$

anonymous · Answer

(10-4sqrt 3)  +  i(8+ 5 sqrt 3)

anonymous · Answer

how did you get 10-4sqrt3?

anonymous · Answer

are you foiling again?

anonymous · Answer

No, I am just arranging the terms into the form a + i b

anonymous · Answer

ok so hold

anonymous · Answer

so if i came up with $$10+8i+5\sqrt{3}i+4\sqrt{3}i ^{2}$$

anonymous · Answer

that's correct, just rearrange now...

anonymous · Answer

how would I know which constant to attach too what radical, could my asnwer be differnt

anonymous · Answer

4 sqrt 3 i^2 = -4 sqrt 3, a number.


So it goes with 10, also a number.

anonymous · Answer

oh i get it

anonymous · Answer

and with 5rad 3 I, I can just put the I to the side since its not under the radical sign?

anonymous · Answer

Correct

anonymous · Answer

thank you for your help

anonymous · Answer

ur welcome.