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

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We have to simplify [(a+i)^3-(a-i)^3]/[(a+i)^2-(a-i)^2].

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

we open the brackets

=> [a^3 + 3a^2*i+ 3a*i^2+ i^3 - a^3 + 3*a^2*i - 3*a*i^2 + i^3]/[(a^2 + 2ai + i^2 - a^2 - i^2 + 2ai]

cancel the common terms in the numerator and denominator

=> [ 3a^2*i+ i^3  + 3*a^2*i + i^3]/[2ai + 2ai]

=> [6a^2*i+ 2i^3 ]/[4ai]

=> [3a^2*i+ i^3 ]/[2ai]

=> [3a^2*i - i ]/[2ai]

=> [3a^2 - 1 ]/[2a]

=> ( 3a^2 - 1)/ 2a

The required simplified form is ( 3a^2 - 1)/ 2a

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