If (x + iy)^1/3 = a + ib, then x/a+y/b =

Question

If (x + iy)^1/3 = a + ib, then \(\frac{x}{a}+\frac{y}{b}\) =

A. 0 

B. 1 

C. -1 

D. None of these

Answer 1

x + iy)^1/3 = a + ib 

x + iy = (a + ib)³ 

= a³ + (ib)³ + 3a²(ib) + 3a(ib)² 

= a³ – ib³ + i3a²b – 3ab² 

= (a³ – 3ab²) + i(3a²b – b3) 

x = a³ – 3ab² and y = 3a²b – b³

\(\frac{x}{a}+\frac{y}{b}\)  = \(\frac{a^3-3ab^2}{a}+\frac{3a^2b-b^3}{b}\)

= a³ – 3b² + 3a² – b³

= 4(a² – b²)