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If  $z=x+iy, z^{1/3} = a - ib,$ then $\frac{x}{a}-\frac{y}{b} = k(a^{2}-b^{2})$ where k is equal to 

Answer:

Explanation:

$z^{1/3} = a-ib\Rightarrow z = (a -ib)^{3}$

$x +iy = a^{3} + ib^{3} - 3ia^{2}b -3ab^{2}$

$Then x= a^{3} -3ab^{2}\Rightarrow \frac{x}{a} = a^{2}-3b^{2}$

$y =b^{3} -3a^{2}b\Rightarrow \frac{y}{b} = b^{2}-3a^{2}$

$So, \frac{x}{a}-\frac{y}{b} = 4(a^{2}-b^{2})$