[tex] \sqrt{ \frac{-49}{(7-2i)-(4+9i) } }
[/tex]
This one is quite the deal, but we can begin by distributing the negative on the denominator and getting rid of the parenthesis:
[tex] \frac{ \sqrt{-49}}{7-2i-4-9i} [/tex]
See how the denominator now is more a simplification of like terms, with this I mean that you operate the numbers with an "i" together and the ones that do not have an "i" together as well. Namely, the 7 and the -4, the -2i with the -9i.
Therefore having the result:
[tex] \frac{ \sqrt{-49} }{3-11i} [/tex]
Now, the [tex] \sqrt{-49} [/tex] must be respresented as an imaginary number, and using the multiplication of radicals, we can simplify it to [tex] \sqrt{49} \sqrt{-1} [/tex]
This means that we get the result 7i for the numerator.
[tex] \frac{7i}{3-11i} [/tex]
In order to rationalize this fraction even further, we have to remember an identity from the previous algebra classes, namely: [tex]x^2 - y^2 =(x+y)(x-y)[/tex]
The difference of squares allows us to remove the imaginary part of this fraction, leaving us with a real number, hopefully, on the denominator.
[tex] \frac{7i (3+11i)}{(3-11i)(3+11i)} [/tex]
See, all I did there was multiply both numerator and denominator with (3+11i) so I could complete the difference of squares.
See how [tex](3-11i)(3+11i)= 3^2 -(11i)^2 [/tex] therefore, we can finally write:
[tex] \frac{7i(3+11i)}{3^2 - (11i)^2 } [/tex]
I'll let you take it from here, all you have to do is simplify it further.
The simplification is quite straightforward, the numerator distributed the 7i. Namely the product [tex]7i(3+11i) = 21i+77i^2 [/tex].
You should know from your classes that i^2 = -1, thefore the numerator simplifies to [tex]-77+21i[/tex]
You can do it as a curious thing, but simplifying yields the result:
[tex] \frac{-77+21i}{130} [/tex]
