Question

Rationalize the denominator of fraction with numerator square root of negative 36 and denominator the quantity of 2 minus 3i plus the quantity 3 plus 2i. @phi

Answer 1

\[ \frac{\sqrt{-36}}{2-3i+3+2i} \]
?

Answer 2

yes and 2-3i is in parenthesis, so is 3+2i

Answer 3

first, I assume you can simplify sqrt(-36) ?

Answer 4

yes

Answer 5

what is it ?

Answer 6

would it be 6i/(2-3i)+(3+2i)

Answer 7

yes, next in the bottom add real to real and imaginary to imaginary
what do you get for the bottom ?

Answer 8

5-i

Answer 9

so now we have
\[ \frac{6i}{5-i}\]
multiply top and bottom by the complex conjugate of 5-i
do you know how to do that ?

Answer 10

idk

Answer 11

the complex conjugate of 5-i is 5+i (you "switch" the sign of the imaginary part)
we now do
(5-i)(5+i)

notice this is similar to (a+b)(a-b) = a^2 - b^2 (we can use FOIL to show this is what you get)

can you do
(5-i)(5+i)
?

Answer 12

6+30i/25

Answer 13

can you do (5-i)(5+i) ?

Answer 14

26

Answer 15

yes it is 5^2 - i^2 or 25 - -1 = 25+1 = 26
if you do a lot of these you remember to just square both numbers and add:
25+1

so you have
\[ \frac{6i(5+i)}{26} \]
next "distributre" the 6i (multiply each term inside by 6i)

Answer 16

-6+30i

Answer 17

so you have
\[ \frac{-6+30i}{26} \]
it looks like we can divided top and bottom by 2 and simplify to get
\[ \frac{-3+15i}{13} \]
or
\[ -\frac{3}{13} + \frac{15}{13} i \]

Answer 18

thank you