SOLUTION: Q.1 If {{{ (x+iy)^(1/3)=a+ib }}} then show that {{{ 4(a^2 - b^2)=x/a + y/b }}} Q.2 Find the real no.s x and y if {{{ (x-iy)(3+5i) }}} is the conjugate of {{{ -6-24i }}} .

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Q.1 If {{{ (x+iy)^(1/3)=a+ib }}} then show that {{{ 4(a^2 - b^2)=x/a + y/b }}} Q.2 Find the real no.s x and y if {{{ (x-iy)(3+5i) }}} is the conjugate of {{{ -6-24i }}} .      Log On


   

Question 1060987: Q.1 If +%28x%2Biy%29%5E%281%2F3%29=a%2Bib+ then show that +4%28a%5E2+-+b%5E2%29=x%2Fa+%2B+y%2Fb+
Q.2 Find the real no.s x and y if +%28x-iy%29%283%2B5i%29+ is the conjugate of +-6-24i+ .
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Q.1 %28x%2Biy%29%5E%28%221+%2F+3%22%29=a%2Bib <---> x%2Biy=%28a%2Bib%29%5E3
x%2Biy=%28a%2Bib%29%5E3
x%2Biy=a%5E3%2B3a%5E2%2A%28ib%29%2B3a%2A%28ib%29%5E2%2B%28ib%29%5E3
x%2Biy=a%5E3%2B3ba%5E2%2Ai%2B3a%2A%28-b%5E2%29-b%5E3%2Ai
x%2Biy=%28a%5E3-3ab%5E2%29%2B%283ba%5E2-b%5E3%29%2Ai
x%2Biy=a%28a%5E2-3b%5E2%29%2Bb%283a%5E2-b%5E2%29%2Ai ---> system%28x=a%28a%5E2-3b%5E2%29%2Cy=b%283a%5E2-b%5E2%29%29 ---> system%28x%2Fa=a%5E2-3b%5E2%2Cy%2Fb=3a%5E2-b%5E2%29%29
So,
x%2Fa%2By%2Fb=a%5E2-3b%5E2%2B3a%5E2-b%5E2
x%2Fa%2By%2Fb=4a%5E2-4b%5E2
x%2Fa%2By%2Fb=4%28a%5E2-b%5E2%29

Q.2 %28x-iy%29%283%2B5i%29=3x%2B5x%2Ai-3y%2Ai%2B5y=%283x%2B5y%29%2B%285x-3y%29%2Ai
The conjugate of -6-24i is -6%2B24i .
%283x%2B5y%29%2B%285x-3y%29%2Ai=-6%2B24i --> system%283x%2B5y=-6%2C5x-3y=24%29 --> highlight%28system%28x=3%2Cy=-3%29%29
There are many ways to solve the system. For example,
we could start by multiplying the first equation times (-6 , the second equation times 3 and adding them to get
-15x-25y%2B15x-9y=30%2B72 --> -34y=102 --> y=102%2F%28-34%29 --> y=-3%29 .
Then, substituting -3 for x in 3x%2B5y=-6, we get
3x%2B5%28-3%29=-6 --> 3x-15=-6 --> 3x=15-6 --> 3x=9 --> x=9%2F3 --> x=3 .