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Question
If `x^4 + 1/x^4 = 194, "find" x^3 + 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^4 + 1/x^4 = 194`
We know that,
`(x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2` ...[Using (a + b)2 = a2 + b2 + 2ab]
Now, substituting the given value
`(x^2 + 1/x^2)^2 = 194 + 2`
∴ `(x^2 + 1/x^2)^2 = 196`
∴ `x^2 + 1/x^2 = sqrt196`
∴ `x^2 + 1/x^2 = +-14`
Let’s relate it to `x^3 + 1/x^3`
`(x + 1/x)^2 = x^2 + 1/x^2 + 2` ...[Using (a + b)2 = a2 + b2 + 2ab]
`(x + 1/x)^2 = 14 + 2`
∴ `x + 1/x = sqrt16`
∴ `x + 1/x = +-sqrt4`
Thus,
`x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x)`
`x^3 + 1/x^3 = (4)^3 - 3(4)`
`x^3 + 1/x^3 = 64 - 12`
∴ `x^3 + 1/x^3 = +-52`
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