Question

Prove that root 2 + root 5 is irrational



Imp
CBSE important question

Answer 1

Answer:

√2 + √5 is an irrational number.

Explanation:

Let √2+√5 be a rational number.

A rational number can be written in the form of ( p / q )  p,q are integers and q ≠  0.

√2+√5 = p/q

Squaring on both sides,

( √2 + √5 )^2 = ( p / q )^2

√2^2 + √5^2 + 2 ( √5 ) . ( √2 ) = 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

2√10 = ( p^2 - 7q^2 ) - q^2

√10 = ( p^2 - 7q^2 ) - 2q^2

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

Then √10 is also a rational number.

But this contradicts the fact that √10 is an irrational number.

∴  Our supposition is false. ( Proof by contradiction )

√2 + √5 is an irrational number.

Hence proved.

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