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Irrational to the Fraction

Let \(a\) be an irrational number and let \(n\) be an integer greater than 1. Prove that

\((a + \sqrt{a^2 - 1})^{\frac {1}{n}} + (a - \sqrt{a^2 - 1})^{\frac {1}{n}}\)

is an irrational number.

Note by Sharky Kesa
3 years, 8 months ago

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Hint: Assume a quadratic :

\(x^{2}-2ax+1\).

With roots as:

\( a+ \sqrt{a^{2}-1}\) & \(a- \sqrt{a^{2}-1}\)

And then use this and try to get the expression above...

Vishal Sharma - 3 years, 8 months ago

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