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Question
If \[x - \frac{1}{x} = 7\], find the value of \[x^3 - \frac{1}{x^3}\].
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Solution
In the given problem, we have to find the value of `x^3 - 1/x^3`
Given: `x - 1/x = 7`
We shall use the identity (a – b)3 = a3 – b3 – 3ab(a – b)
Here putting, `x - 1/x = 7`,
`(x - 1/x)^3 = x^3 - 1/x^3 - 3 (x xx 1/x)(x - 1/x)`
`(7)^3 = x^3 - 1/x^3 - 3 (x xx 1/x ) (x-1/x)`
`343 = x^3 - 1/x^3 - 3 (x - 1/x)`
`343 = x^3 - 1/x^3 - 3 xx 7 `
`343 = x^3 - 1/x^3 - 21`
`343 + 21 = x^3 - 1/x^3`
`343 = x^3 - 1/x^3`
Hence the value of `x^3 - 1/x^3` is 364.
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