Q.222. Algebra.

Question 222.
If x^4 + 1/x^4 = A, then what is x^5 + 1/x^5 in terms of A?
What is its positive, real value if A = 47?

Answer 222.
(x^2 + 1/x^2)^2 - 2 = A
=> x^2 + 1/x^2 = sqrt(A + 2)
=> (x + 1/x)^2 - 2 = sqrt(A + 2)
=> x + 1/x = sqrt [sqrt(A + 2) + 2]

x^5 + 1/x^5
= (x + 1/x) (x^4 - x^2 + 1 - 1/x^2 + 1/x^4)
= sqrt [sqrt(A + 2) + 2] * [A + 1 - sqrt(A + 2)]

For A = 47
x^5 + 1/x^5
= sqrt [sqrt(47 + 2) + 2] * [47 + 1 - sqrt(47 + 2)]
= 123.

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