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Write the value of 25^{3} − 75^{3 }+ 50^{3}.

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#### Solution

The given expression is

`25^3 -75^3 + 50^3`

Let a =25, b =-75 and c = 50. Then the given expression becomes

`25^3 - 75 + 50^3 = a^3 +b^3 + c^3`

Note that

`a+b+c = 25 + (-75) + 50`

` = 25 - 75 + 50`

`= 0`

Recall the formula

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

When a + b + c = 0 , this becomes

`a^3 + b^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`

`25^3 - 75^3 + 50^3 = 3.(25).(-75).(50)`

`25^3 - 75^3 + 50^3 = -281250`

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