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प्रश्न
If `x + (1)/x = 5`, find the value of `x^2 + (1)/x^2, x^3 + (1)/x^3` and `x^4 + (1)/x^4`.
उत्तर
`x + (1)/x = 5` ...(1)
Squaring both sides of (1)
`(x + 1/x)^2` = (5)2
⇒ `x^2 + (1)/x^2 + 2` = 25
⇒ `x^2 + (1)/x^2`
= 25 - 2
= 23 ...(2)
Cubing both sides of (1),
`(x + 1/x)^3` = 953
`x^3 + (1)/x^3 + 3 (x + 1/x)` = 125
⇒ `x^3 + (1)/x^3 + 3 (5)` = 125
⇒ `x^3 + (1)/x^3`
= 125 - 15
= 110
Squaring both sides of (2),
`(x^2 + 1/x^2)^2` = (23)2
⇒ `x^4 = (1)/x^4 + = 529`
⇒ `x^4 + (1)/x^4`
= 529 - 2
= 527.
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