Roots of Complex Numbers (Sharp, TI, Casio)

[Matt Agajanian]

Hi all.

On a whim (and maybe a hunch, but a damn lucky guess), I keyed in a method that calculates n-th roots of Complex Numbers on the
TI-36X Pro/30X Pro MathPrint
Sharp EL-W516X and T
Casio 115ES and 991 line.

It goes like this:

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

General algorithm:
Calculate n-th root of x+yi

[math] [P→Rx] (x^(1/n),y/n) → a [Real part]
[math] [P→Ry] (x^(1/n),y/n) → b [Imaginary part]

Casio & Sharp models would then follow the above algorithm.

But then I thought maybe the algorithm for Raising i to integer powers could be modified as an alternative method.

For a reminder,

Example 1: (11 + 2i)^4 = 11753 + 10296i

(Radian Mode)
R>Pr(11,2)^4 sto→ x (15625)
R>PΘ(11,2)*4 sto→ y (0.719413999)
P>Rx(x,y) returns 11753
P>Ry(x,y) returns 10296

So, wouldn't just switching to this:

Example: Calculate the 4th root of a+bi
(Radian Mode)
R>Pr(a,b)^(1/4) sto x
R>PΘ(a,b)/4 sto y
P>Rx(x,y)
P>Ry(x,y)

or would this work:

(Radian Mode)
R>Pr(a^(1/4),b/4) sto x
R>PΘ(a^(1/4),b/4) sto y
P>Rx(x,y)
P>Ry(x,y)
?

or the first algorithm noted above the accurate and only method?