If x, y, a, b are real numbers such that (x+i y)^ 1 / 5 =a+i b and p= x a - y b , then

Step-by-step solution

aligned & (x+i y)^ 1 5 =a+i b & x+i y=(a+i b)^5=a^5+5 a^4(i b)+10 a^3 (-b^2 )+10 a^2 (-i b^3 )+5 a b^4+i b^5 aligned Equating the real and imaginary parts x=a^5-10 a^3 b^2+5 a b^4 y=5 a^4 b-10 a^2 b^3+b^5p= x a - y b = (a^4-10 a^2 b^2+5 b^4 )- (5 a^4-10 a^2 b^2+b^4 )=-4 a^4+4 b^4=-4 (a^4-b^4 ) aligned & =-4 (a^2-b^2 ) (a^2+b^2 )=-4(a+b)(a-b)(a+i b)(a-i b) & a-b, a+b, a+i b and a-i b are all factors of p . aligned