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Show that 5 + √7 is an irrational number.

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Let us assume that 5 + √7 is a rational number. 

So, we can find co-prime integers ‘a’ and ‘b’ (b ≠ 0) such that

 \(5+\sqrt7 = \frac{a}{b}\) 

\(\therefore \sqrt7 = \frac{a}{b} - 5\) 

Since, ‘a’ and ‘b’ are integers,\( \sqrt[a]b \) – 5 is a rational number and so √7 is a rational number. 

∴ But this contradicts the fact that √7 is an irrational number. 

Our assumption that 5 + √7 is a rational number is wrong. 

∴ 5 + √7 is an irrational number.

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asked Aug 28, 2018 in Mathematics by Mubarak (33.0k points)

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