What is the twenty seventh power of an imaginary number?

Answer 1

See the Wikipedia article on Imaginary Numbers. i^n = i^(n mod 4). With n = 27, 27 mod 4 = 3, and i^3 = -i. This is easier to visualize when you consider the graphical representation of complex numbers, and use polar coordinates. Writing i as exp(i*pi/2), (from Euler's formula), then i^27 = {using exp() to mean the natural base e, raised to a power} exp(i*pi/2)^27 = exp(27*i*pi/2) = exp(13.5*i*pi) = exp((12 + 1.5)*(i*pi)) = exp(12*i*pi)*exp(3*i*pi/2).

Since the coefficient of i in the exponent is an angle (in radians), then even multiples of pi are the same angle as 0 {exp(0) = 1} so we are back to the same as exp(3*i*pi/2), which is pointing straight down [-i]. Note that 3*pi/2 radians is the same as 270°.

Since the question asked about 27th power of an imaginary number, that could mean a multiple of i, such as bi, where b is any real number. In this case, you would have (bi)^27 = (b^27)(i^27) = (b^27)(-i). So if b = 1.5 for example, then you would have (-i)(1.5^27) ≅ -56815i.