Roots of Complex Numbers (Sharp, TI, Casio)

[Albert Chan]

z = |z| * e^(θ*i), where θ = arg(z)

(a^b)^c = a^(b*c)      → z^k = |z|^k * e^(kθ*i)

(12-30-2022 10:58 PM)Matt Agajanian Wrote:  Example Calculate 4th root of (15625+0.719413999i)

Let x= 15625, y=0.719413999

TI-30X Pro MathPrint:
[math] [P→Rx] (x^.25,y/4) → 11 [Real part]
[math] [P→Ry] (x^.25,y/4) → 2 [Imaginary part]
Thus, the answer is 11+2i

This example is in error, since (11 + 2i)^4 = 11753 + 10296i
This is what it mean:

(15625 * e^(0.719413999i)) ^ 0.25
= (15625 ^ 0.25) * e^(0.719413999i * 0.25)
≈ (11 + 2i)