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Question
If `( a + 1/a )^2 = 3 "and a ≠ 0; then show:" a^3 + 1/a^3 = 0`.
Sum
Solution
Given that `( a + 1/a )^2 = 3`
⇒ `a + 1/a = +- sqrt3` ...(1)
We need to find `a^3 + 1/a^3`
Consider the identity,
`( a + 1/a )^3 = a^3 + 1/a^3 + 3( a + 1/a )`
⇒ `a^3 + 1/a^3 = ( +- sqrt3 )^3 - 3( +-sqrt3 )` ...[From (1)]
⇒ `a^3 + 1/a^3 = +-3sqrt3 - 3(+- sqrt3 )`
⇒ `a^3 + 1/a^3 = 0`
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