Question

prove that √2+√5 is irrational​

Answer 1

Given: √2+√5

We need to prove√2+√5 is an irrational number.

Proof

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

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

√2+√5 = p/q

On squaring both sides we get,

(√2+√5)² = (p/q)²

√2²+√5²+2(√5)(√2) = p²/q²

2+5+2√10 = p²/q²

7+2√10 = p²/q²

2√10 = p²/q² – 7

√10 = (p²-7q²)/2q

p,q are integers then (p²-7q²)/2q is a rational number.

Then √10 is also a rational number.

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

Our assumption is incorrect

√2+√5 is an irrational number.

Hence proved.

Answer 2

p,q are integers then p square-7q square/2q is a rational number.

Then root10 is also a rational number.

But this contradicts thr fact that root1p is an irrational nunber.

Our assumptiin is incorrect.

so root2+root5 is irrational.

HENCE PROVED.