If (1-4 i)^3=a+i b then the value of a and b is

Step-by-step solution

Given the expression (1-4i)^3 = a + ib . Using the identity (x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 , we substitute x = 1 and y = 4i . (1-4i)^3 = (1)^3 - 3(1)^2(4i) + 3(1)(4i)^2 - (4i)^3 = 1 - 12i + 3(16i^2) - 64i^3 Since i^2 = -1 and i^3 = -i , we have: = 1 - 12i + 3(-16) - 64(-i) = 1 - 12i - 48 + 64i = (1 - 48) + (-12 + 64)i = -47 + 52i Comparing this with a + ib , we get a = -47 and b = 52 . Answer: -47,52