#1**+1 **

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.

Melody Jan 30, 2021

#2**0 **

I will admit that this does not fall out as easily as I had hoped. I do not have an answer yet.

Melody
Jan 30, 2021

#3**0 **

Are there any restrictions on a and b. Like do they have to be real or positive or anything?

Melody
Jan 30, 2021

#4**0 **

yeah i've been trying and failing to solve it lol, this is the entire problem, theres nothing else.

SpaceTsunaml
Jan 30, 2021

#5**0 **

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

SpaceTsunaml
Jan 30, 2021

#6**0 **

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

Melody
Jan 30, 2021

#7**+1 **

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 \(a^2+b^3=9+8=17\)

Melody Jan 30, 2021

#8**+1 **

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...

SpaceTsunaml
Jan 30, 2021

#9**+1 **

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

Guest Jan 30, 2021

#10**+1 **

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

SpaceTsunaml
Jan 30, 2021