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Irrational Decision (Posted on 2015-11-15) Difficulty: 3 of 5
Let A be an irrational number and let N is an integer > 1.

Is N√(A + √(A2-1)) + N√( A - √(A2-1)) always an irrational number?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 6 of 6 |
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.

  Posted by Brian Smith on 2016-07-09 22:55:15
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