Let A be an irrational number and let N is an integer > 1.

Is ^{N}√(A + √(A^{2}-1)) + ^{N}√( A - √(A^{2}-1)) always an irrational number?

Give reasons for your answer.

Let S be the value of the expression.

Let X = [A+sqrt(A^2-1)]^(1/n) and Y = [A-sqrt(A^2-1)]^(1/n). Then X+Y=S. Also note that X*Y = 1.

The sum of squares can be calculated by the identity X^2+Y^2 = (X+Y)^2 - 2XY. Higher powers of sums can be calculated from the identity X^k+Y^k = (X+Y)*(X^(k-1)+Y^(k-1)) - (XY)*(X^(k-2)+Y^(k-2).

After substituting X+Y=S and XY=1, then X^n + Y^n can be expressed as a polynomial function of S. If S is rational then X^n+Y^n is rational.

Assume A is irrational and S is rational. X^n+Y^n can be calculated directly as 2A. But S being rational implies X^n+Y^n is rational. This contradicts the assertion that A is irrational. Therefore if A is irrational then S must also be irrational.