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Question
If \[x^4 + \frac{1}{x^4} = 194,\] then \[x^3 + \frac{1}{x^3} =\]
Options
76
52
64
none of these
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Solution
Given `x^4 +1/x^4 = 194`
Using identity `(a+b)^2 = a^2+2ab+b^2`
Here, `a= x^2 , b = 1/x^2`
`(x^2 +1/x^2 )^2 = (x^2)^2 + 2 xx x^2 xx 1/x^2 +1/(x^2)^2`
`(x^2 + 1/x^2 )^2 = x^4 +1/x^4 +2`
`(x^2+1/x^2)^2 = 194 +2`
`(x^2+1/x^2)^2 = 196`
`(x^2+1/x^2)(x^2+1/x^2)^2 = 14 xx14`
`x^2+1/x^2 = 14`
Again using identity `(a+b)^2 = a^2 +2ab +b^2`
Here `a=x,b=1/x`
`(x+1/x)^2 = (x)^2 + 2 xx x xx 1/x +1/(x)^2`
`(x+1/x)^2 = x^2 + 2 + 1/x^2`
Substituting `x^2 +1/x^2 = 14`
`(x+1/x)^2 = 14 +2`
`(x+1/x)^2 = 16`
`x+1/x = 4`
Using identity `a^3 +b^3 = (a+b)(a^2 - ab +b^2)`
Here `a= x^3, b= 1/x^3`
`x^3 +1/x^3 = (x+1/x)(x^2 - x xx 1/x+1/x^2)`
`x^3 +1/x^3 = (4)(-1 +14)`
`x^3 +1/x^3 = (4)(13)`
`x^3 +1/x^3 = 52`
Hence the value of `x^3 +1/x^3`is 52.
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