If (x-i y)^ 1 / 3 -a-i b , then the value of x 2 a + y 2 b is

Step-by-step solution

(x-i y)^ 1 / 3 =a-i b(x-i y)=(a-i b)^3 aligned & =a^3-i^3 b^3-3 a^2( ib )+3 a ( ib )^2 & =a^3+i b^3-3 a^2 b i-3 a b^2 aligned aligned & x-i y= (a^3-3 a b^2 )+i b (b^2-3 a^2 ) & x=3 a^3-3 a b^2 aligned x a =a^2-3 b^2 x 2 a = a^2-3 b^2 2 ...(i) and y=b (3 a^2-b^2 ) y 2 b = 3 a^2-b^2 2 ...(ii) y=b (3 a^2-b^2 ) (i) + (ii) x 2 a + y 2 b = a^2-3 b^2+3 a^2-b^2 2 aligned & = 4 a^2-4 b^2 2 & =2 (a^2-b^2 ) aligned