Question

Write the value of 48³ − 30³ − 18³.

Answer 1

The given expression is

 `48^3 - 30^3 -18`

Let a = 48, b =  - 30 and  c = -18 . Then the given expression becomes

`48^3 - 30^3 - 18^3 = a^3 + b^3 + c^3`

Note that

`a+b+c = 48+ (-30) + (- 18)`

                 ` = 48 - 30 - 18`

                 `= 0`

Recall the formula `a^2 + b^3 + c^3-3abc = (a+b+c)(a^2 + c^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`

`48^3 - 30^3 - 18^3 = 3.(48).(-30).(-18)`

`48^3 -30^3 - 18^3 = 77760`