Simplify the fraction [(a+i)^3-(a-i)^3]/[(a+i)^2-(a-i)^2]

We'll re-write the difference of cubes from
numerator:

(a+i)^3-(a-i)^3 = (a + i - a + i)[(a+i)^2 +
(a+i)(a-i) + (a-i)^2]

We'll combine and eliminate like
terms:

(a+i)^3-(a-i)^3 = (2 i)(a^2 + 2ai + i^2 + a^2 - i^2
+ a^2 - 2ai + i^2)

(a+i)^3-(a-i)^3 = (2 i)(3a^2 +
i^2)

(a+i)^3-(a-i)^3 = (2 i)(3a^2
-1)

We'll re-write the difference of squares from
numerator:

[(a+i)^2-(a-i)^2] = (a + i - a + i)(a + i + a -
i)

We'll combine and eliminate like
terms:

[(a+i)^2-(a-i)^2] =
2i*2a

We'll re-write the
fraction:

 (2 i)(3a^2 -1)/2i*2a = (3a^2
-1)/2a

3a^2 -1  = (a*sqrt3 - 1)(a*sqrt3 +
1)

[(a+i)^3-(a-i)^3]/[(a+i)^2-(a-i)^2] =
[(a*sqrt3 - 1)(a*sqrt3 + 1)]/2a