Expert: Josh - 3/22/2004

Question
hi
if a= -4(under root 3)
than a2? ( read: a square)

Answer
Hi Ranish,

There are various ways of doing this using complex numbers.
You can define i=sqrt(-1).
In general, a complex number, z, may be represented as z=x+iy, where x,y are actual real numbers.
x=Re(z) is referred to as the real part,
y=Im(z) is referred to as the imaginary part for obvious reasons.

So, we can write i=cos(pi/2)+i*sin(pi/2) in the cartesian form z=x+iy.

If you write "i" in polar form, mod(z)/_arg(z),
where modulus mod(z)=sqrt(x^2+y^2),
     principal argument (angle in Argand plane)
     arg(z)=atan(y/x) in [-pi,pi].

i=1 /_ (j*pi/2). Sometimes, we write i=e(j*pi/2) for short.

Then, a=(-4)^(1/3)=(4*i^2)^(1/3).
Using index law,
a=4^(1/3)*[cos(pi/3)+i*sin(pi/3)];
since i^2=e(j*pi) and (i^2)^(1/3)= 1 /_ (j*pi*1/3)

Therefore, a=4^(1/3)*[1/2+i*sqrt(3)/2]