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Question
If \[x^2 + \frac{1}{x^2}\], 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^2 + 1/x^3 = 51`
We shall use the identity `(x+y)^2 = x^2 + y^2 +2xy`
Here putting `x^2 + 1/x^3 = 51`,
`(x - 1/x)^2 = x^2 +1/x^2 - 2 xx x xx 1/x`
`(x - 1/x)^2 = x^2 +1/x^2 - 2 xx x xx 1/x`
`(x - 1/x)^2 = 51 - 2`
`(x - 1/x)^2 = 49`
`(x - 1/x) = sqrt49`
`(x - 1/x) =±7`
In order to find `x^3 - 1/x^3`we are using identity `a^3 - b^3 = (a-b)(a^2 +b^2 +ab)`
`x^3 - 1/x^3 = (x-1/x)(x^2 + 1/x^2 + x xx 1/x)`
`x^3 - 1/x^3 = (x-1/x)(x^2 + 1/x^2 + x xx 1/x)`
Here `(x-1/x)= 7`and `x^2 + 1/x^2 = 51`
`= 7 (51 +1)`
` = 7 xx 52`
` = 364`
Hence the value of `x^3 - 1/x^3`is 364.
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