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Question
If \[x - \frac{1}{x} = 7\] ,find the value of \[x^3 - \frac{1}{x^3}\]
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 = a^3 - b^3 - 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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