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If \[x - \frac{1}{x} = 5\], find the value of \[x^3 - \frac{1}{x^3}\]
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In the given problem, we have to find the value of `x^3 - 1/x^3`,
Given: `x - 1/x = 5`,
We shall use the identity (a − b)3 = a3 − b3 − 3ab(a − b),
Here putting, `x- 1/x = 5`,
`(x-1/x)^3 = x^3 - 1/x^3 - 3(x xx1/x)(x - 1/x)`
`(5)^3 = x^3 - 1/x^3 - 3(x xx1/x)(x - 1/x)`
`125 = x^3 - 1/x^3 - 3(x - 1/x)`
`125 = x^3 - 1/x^3 - 3xx5`
`125 = x^3 - 1/x^3 - 15`
`125 + 15 = x^3 - 1/x^3`
`140 = x^3 - 1/x^3`
Hence, the value of `x^3 - 1/x^3` is 140.
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