Question

Prove that 2 + root 5 is irrational

Answer 1

hey dear


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here is your answer


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Let √2 + √5 be rational number



A rational number can be written in the form of p / q integer unequal to 0


√2 + √5 = p / q



squaring on both the sides we get



( √2 + √5 ) ^2 = ( p / q ) ^2




( √2 )^2 + ( √5 ) ^2 + 2 ( √2 ) (√5) = p^2 / q ^2


2 + 5 +2 √10 = p ^2. / q ^2


7 + 2√10 = p^2. / q ^2.


2√10 = p^2 / q ^2. - 7



√10. = ( p^2 - 7q^2 ) / 2



p, q are integers then ( p^2 - 7q^2 ) / 2. are rational number


so √10 is also a rational number



but it contradicts our facts that √10 is a rational number



so, our supposition is false



hence √2 +√5 is irrational number


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hope it helps


thank you

Answer 2

$\huge \boxed{ \underline{ \underline{ \bf{Answer}}}}$

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Let us assume that√2 + √5 is a rational number.

Then, there exist coprime positive integers a and b such that :-

√2 + √5 = a/b

a/b - √2 = √5

(a/b-√2)² = (√5)²   ( squaring both sides )

a²/b² - 2a/b√2 + 2 = 5

a²/b²- 3 = 2a/b√2

a²-3b²/2ab = √2

√2 is a rational number.

[ a,b are integers ; a²-3b²/2ab is rational ]

This contradicts the fact that √2 is irrational number. Our assumption is wrong.

Hence, √2 + √5 is irrational !

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