Question

Write the value of \[\left( \frac{1}{2} \right)^3 + \left( \frac{1}{3} \right)^3 - \left( \frac{5}{6} \right)^3 .\]

Answer 1

The given expression is

 `(1/2)^3 + (1/3)^3 - (5/6)^3`

Let,  `a=1/2 ,b=1/3`and. `c = -5/6`Then the given expression becomes

 `(1/2)^3 + (1/3)^3 - (5/6)^3  = a^3 + b^3+c^3`

Note that:

`a+b+c = 1/2 + 1/3 + (-5/6)`

                 ` = 1/2 + 1/3 - 5/6`

                 ` =0`

Recall the formula

 `a^3 +b^3 + c^3 - 3abc - 3abc = (a+b+c)(a^2 +b^2 + c^2 - ab - bc - ca)`

When a + b +  c = 0, this becomes

`a^3 +b^3 + c^3 - 3abc = 0. (a^2 +b^2 + c^2 - ab - bc - ca)`

                                   ` =0 `

             `a^3 + b^3 + c^3  = 3abc`

So, we have the new formula

 `a^3 + b^3 + c^3 = 3abc`, when a + b + c = 0.

Using the above formula, the value of the given expression is

                        `a^3 +b^3 +c^3 = 3abc`

`(1/2)^3 + (1/3)^3 - (5/6)^3 = 3.(1/2).(1/3).(-5/6)`

    `1/2 ^3 + (1/3)^3  - (5/6)^3 = - 5/12`