Question

Prove that root 2 + root 5 is irrational number

Answer 1

Answer:

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

Thus,we can find coprime integers a and b such that √2+√5=a/b where b not =0

√2+√5=a/b

√5=a/b-√2

√5=a-√2b/b

(√5)^2={a-√2/b}^2

5=a^2/b^2+2-2√2a/b

3-a^2/b^2=-2√2a/b

-3b^2+a^2/b^2×b/a=2√2

-3b^2+a^2/ab=2√2

-3b^2+a^2/2ab=√2

since LHS is a rational number so RHS is also a rational number.

So,√2 is a rational number but this contradict the fact that √2 is an irrational no.

This contradiction has arisen due to our wrong assumption that √2+√5 is a rational number.

So, we concluded that √2+√5 is an irrational no.

Hence,proved.

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