Question

prove that root 2+root 5 is irrational
$\sqrt{2 + \sqrt{5} }$
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Answer 1

Given that √5 is irrational

Let √(2 + √5) be a rational number,

Therefore, It can be expressed in the form of p/q where p&q are co-prime integers.

p/q = √( 2 + √5)

Square both sides

p² / q² = (√(2+√5))²

p² /q² = 2 + √5

p² /q² - 2 = √5

p² - 2q² / q² = √5

Here, (p² - 2q²)/q² is a rational number but √5 is an irrational Since an irrational cant be equal to a rational number So our assumption is wrong and Hence,

√(2 + √5) is an irrational number.