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16 Aug 2007, 04:11
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If X+1/X = 4, find the value of X^4 - 1/x^4(x>1)
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16 Aug 2007, 07:57
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X^4 - 1/x^4 = ?
we can write it like
(X^2 + 1/x^2)(X^2 - 1/x^2)
1st root (X^2 + 1/x^2)
2nd root (X^2 - 1/x^2)
Soln of 1st root
x + 1/x =4
Now square it
(x + 1/x)^2 =4^2
=>
x^2 + 1/x^2 + 2 = 16
=>
x^2 + 1/x^2 = 14
Soln of 2nd root
(X^2 - 1/x^2)
square it
(X^2 - 1/x^2)^2 =x^4 + 1/x^4 -2 ....... let this is euation 1
Now square the soln of 1st root
(x^2 + 1/x^2)^2 =14^2
=>
x^4 + 1/x^4 + 2 = 196
=>
x^4 + 1/x^4 = 194
Now keep this value into equation 1
(X^2 - 1/x^2)^2 =x^4 + 1/x^4 -2 = 194 - 2 = 192
=> (X^2 - 1/x^2) =sqrt(192) =8 sqrt(3)
Now we got both root
multiple both
(X^2 + 1/x^2)(X^2 - 1/x^2) = 14 * 8 * sqrt(3)
we can write it like
(X^2 + 1/x^2)(X^2 - 1/x^2)
1st root (X^2 + 1/x^2)
2nd root (X^2 - 1/x^2)
Soln of 1st root
x + 1/x =4
Now square it
(x + 1/x)^2 =4^2
=>
x^2 + 1/x^2 + 2 = 16
=>
x^2 + 1/x^2 = 14
Soln of 2nd root
(X^2 - 1/x^2)
square it
(X^2 - 1/x^2)^2 =x^4 + 1/x^4 -2 ....... let this is euation 1
Now square the soln of 1st root
(x^2 + 1/x^2)^2 =14^2
=>
x^4 + 1/x^4 + 2 = 196
=>
x^4 + 1/x^4 = 194
Now keep this value into equation 1
(X^2 - 1/x^2)^2 =x^4 + 1/x^4 -2 = 194 - 2 = 192
=> (X^2 - 1/x^2) =sqrt(192) =8 sqrt(3)
Now we got both root
multiple both
(X^2 + 1/x^2)(X^2 - 1/x^2) = 14 * 8 * sqrt(3)
