Question

Simplify of the following:

\[\left( \frac{x}{2} + \frac{y}{3} \right)^3 - \left( \frac{x}{2} - \frac{y}{3} \right)^3\]

Answer 1

In the given problem, we have to simplify equation 

Given \[\left( \frac{x}{2} + \frac{y}{3} \right)^3 - \left( \frac{x}{2} - \frac{y}{3} \right)^3\]

We shall use the identity  `a^3 - b^3 = (a-b)(a^2+b^2 + ab)`

Here `a=(x/2 + y/3 ),b= (x/2 - y/3)`

By applying identity we get 

`((x/2 +y/3) -(x/2 - y/3)) [(x/2 +y/3)^2 + (x/2 - y/3)^2 - (x/2 +y/3) (x/2 -y/3)  ]`

` = (x/2 + y/3 - x/2+y/3) [((x/2)^2+(y/3)^2 + (2xy)/6)^2 + ((x/2)^2+ (y/3)^2  - (2xy)/6)^2 + ((x/2)^2 - (y/3)^2) )]`

`= (2y)/3 [(x^2 /4 + y^2/9 +(2xy)/6)  + (x^2/4 + y^2/9 - (2xy)/6) + x^2/4 - y^2/9]`

` =( 2y)/3 [x^2 /4+ y^2/9 + (2xy)/6 + x^2/4 - y^2/9 - (2xy)/6 + x^2 /4 - y^2/9]`

By rearranging the variable we get

` = (2y)/3 [x^2/4 + y^2/9 + x^2/4 + x^2/4]`

` = (2y)/3 [(3x^2)/4 + y^2/9]`

` = (x^2y)/2 + (2y^3)/27`

Hence the simplified value of`(x/2 + y/3)^3 - (x/2 - y/3)^3` is `(x^2y)/2+(2y^3)/27`