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Question
If `a^2 + 1/a^2` = 18; a ≠ 0 find :
(i) `a - 1/a`
(ii) `a^3 - 1/a^3`
Solution
`a^2 + 1/a^2` = 18`
`( a - 1/a )^2 = a^2 + 1/a^2 - 2`
⇒ `( a - 1/a )^2 = 18 - 2`
⇒ `( a - 1/a)^2 = 16`
⇒ `a - 1/a = +- sqrt16`
⇒ `a - 1/a = +- 4` ...(1)
(ii) `( 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 = (+- 4)^3 + 3(+- 4)` [ From(1) ]
⇒ `a^3 - 1/a^3 = +- 76`
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