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