+118667 I have not finished this but this is how I would start.
\(x=\frac{a+5i}{1-2bi}\\ x=\frac{a+5i}{1-2bi}\times \frac{1+2bi}{1+2bi}\\ x=\frac{(a+5i)(1+2bi)}{(1-2bi)(1+2bi)}\\ etc\\ x \;\;also\;\;=\sqrt2+\frac{3\pi}{4}i\)
Equate coefficients and solve simultaneously.
+118667 I will admit that this does not fall out as easily as I had hoped. I do not have an answer yet.
+118667 Are there any restrictions on a and b. Like do they have to be real or positive or anything?
+138 yeah i've been trying and failing to solve it lol, this is the entire problem, theres nothing else.
+138 Im not entirely sure how but im 90% sure rather than solving for a and b, what is supposed to be done is simplify it down enough to an a^2 + b^3 state
+118667 Yes I expect you are right.
I just solved it and got the silliest answer ever. (there were multiple places where a careless error was probably)
I got
\(a^2+b^3=\frac{324\pi^4-(4320-27\sqrt2)\pi^3+(12096+540\sqrt2)\pi^2+(15360-1456\sqrt2)\pi+(4096+8000\sqrt2))}{2048}\)
Somehow I do not think that is what they want LOL
+118667 Hang on a tick, is it possible that \((\sqrt2, \frac{3\pi}{4})\) is an absolute value and an angle reference?
If so then the rectangular coordinates would be (-1,1) . That looks a lot easier to deal with!
\(\frac{a+5i}{1-2bi}=-1+i\\ \)
I solved this and got a=3 and b=2
so \(a^2+b^3=9+8=17\)
+138 oh my gosh, that makes a lot more sense! LOL I feel so bad, it looks like you spent so long doing that one hahaha oops...
I would not feel so ashamed. To be fair, the question was not worded properly. There should at least be some indication of the meaning of the coordaintes somewhere because coordinates are rectangular by default. Only someone with stellar pattern recognition, like Melody, would recognize what the true intent of the question is. A victim who stumbles upon this ugly question first time will not have a lousy experience
+138 literally lol. IB Sucks, all of the questions are like this, big props to melody for even recognizing the cartesian rectangular coordinates, after you recognized that it took me 2 minutes to solve it lol. I learned about those ages ago but totally forgot
