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Question
If a2 + `1/a^2 = 47` and a ≠ 0 find :
- `a + 1/a`
- `a^3 + 1/a^3`
Solution
(i) `a + 1/a`
`a^2 + 1/a^2 = 47`
`( a + 1/a )^2 = a^2 + 1/a^2 + 2`
⇒ `( a + 1/a )^2 = 47 + 2`
⇒ `( a + 1/a )^2 = 49`
⇒ `a + 1/a = +- sqrt49`
⇒ `a + 1/a = +- 7` ....(1)
(ii) `a^3 + 1/a^3`
`( a + 1/a )^3 = a^3 + 1/a^3 + 3( a + 1/a )`
⇒ `a^3 + 1/a^3 = ( a + 1/a )^3 - 3( a + 1/a )`
⇒ `a^3 + 1/a^3 = ( +- 7 )^3 - 3( +- 7 )` [ From (1) ]
⇒ `a^3 + 1/a^3 = +- 322`
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