Question

how to prove under root 2 and under root 5 are irrational

Answer 1

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

Therefore it can be represented as

m/p=√2+√5 (where m,p are integers with gcd 1)

m/p=√2+√5/1

Therefore m=√2+√5 & p=1

But √2+√5 is not an integer which contradicts our assumption.

Hence √2+√5 is an irrational number

I hope this will help you
if not then comment me

Answer 2

➡️Let √2+√5 be a rational number.

➡️A rational number can be written in the form of p/q where p,q are integers.

√2+√5 = p/q

➡️Squaring on both sides,

(√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 supposition is false.

➡️√2+√5 is an irrational number.

Hence proved.