If z=x-i y and z^ 1 3 =a+i b , then ( x a + y b ) a^2+b^2 =

Step-by-step solution

Given, aligned z & =x-i y and z^ 1 3 =a+i b z^ 1 / 3 & =a+i b aligned Take cube on both sides, we get array ll (z^ 1 / 3 )^3=(a+i b)^3 & z=a^3+(i b)^3+3 a^2 i b+3 a(i b)^2 & z=a^3+i^3 b^3+3 a^2 i b+3 a b^2 i^2 & z=a^3-i b^3+3 a^2 b i-3 a b^2 & [ i^2=-1 ] z= (a^3-3 a b^2 )-i (b^3-3 a^2 b ) & x-i y= (a^3-3 a b^2 )-i (b^3-3 a^2 b ) [ z=x-i y] array Here, x=a^3-3 a b^2 and y=b^3-3 a^2 b Now, aligned ( x a + y b ) a^2+b^2 & = ( a^3-3 a b^2 a + b^3-3 a^2 b b ) a^2+b^2 & = a^2-3 b^2+b^2-3 a^2 a^2+b^2 & = -2 a^2-2 b^2 a^2+