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Question
Write the value of 483 − 303 − 183.
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Solution
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`
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