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प्रश्न
Write the value of 253 − 753 + 503.
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उत्तर
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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