Rationalize the denominator of sqrt -49 over (7 - 2i) - (4 + 9i)

This one is quite the deal, but we can begin by distributing the negative on the denominator and getting rid of the parenthesis:



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: 



Now, the  must be respresented as an imaginary number, and using the multiplication of radicals, we can simplify it to 
This means that we get the result 7i for the numerator.



In order to rationalize this fraction even further, we have to remember an identity from the previous algebra classes, namely: 
The difference of squares allows us to remove the imaginary part of this fraction, leaving us with a real number, hopefully, on the denominator.



See, all I did there was multiply both numerator and denominator with (3+11i) so I could complete the difference of squares.
See how  therefore, we can finally write:



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 .
You should know from your classes that i^2 = -1, thefore the numerator simplifies to 
You can do it as a curious thing, but simplifying yields the result:

The amount used= 13.04- 1.9 = 11.14