Question

prove that root 5 + root 2 is irrational if p & q are primes​

Answer 1

Answer:

let us assume that √5+√2 is rational

there fore √5+√2=a/b where a and b are co primes b≠0

√5+√2 =a/b

sqauring on both sides

(√5+√2)² =(a/b)²

(√5)²+2(√5)(√2)=a²/b²

5+2√10+2=a²/b²

7+2√10=a²/b²

2√10=a²/b² - 7

2√10=a²/b² -7/1

2√10= a²-7b²/b²

√10=a²-7b²/2b²

irrational≠rational

this contradicts that √5+√2 is rational

therefore our assumption is wrong

so √5+√2 is irrational

Answer 2

Proof:

Let us assume that √2+√5 is rational.

Then √2+√5 =p/q where; p,q € integers and p,q are coprimes.

√2=p/q-√5

By squaring on both sides ,we will get

2=p^2/q^2 +5-3√5p/q

3√5p/q=p^2+2q^2/q^2

√5=p^2+2q^2/2pq

Since p,q are integers p^2+2q^2/2pq is rational.

Therefore, √5 is rational.

which is a contradiction as √5 is rational.

This contraction is due to our assumption.

Thus our assumption is wrong.

HENCE √2+√5 IS RATIONAL.......

Hope you understood my explanation........