Question

can someone please givae an answer
If p and q are irrational then prove that √p+√q is irrational​

Answer 1

Answer:

Here's the answer..

Step-by-step explanation:

lets assume that √p is rational,

⇒ √p = a/b ( where 'a' and 'b' are co primes, meaning they don't have any common factors except for 1)

From squaring both sides,

p = a²/b²

⇒pb² = a²

⇒ b² = a²/p

Since 'p' divides a², it also divides 'a' meaning 'a' has a factor of p

Let 'a' = pm (where m is a positive integer) ⇒ a² = p²m²

Now, pb² = a²

pb² = p²m²

pb²/p²= m²

b²/p =m²

∴ 'p' divides 'b' ⇒ 'b' also has a factor 'p'

∴ 'a' and 'b' are not co primes and our assumption was wrong

⇒ √p is irrational

Similarly √q is irrational

∴⇒ √p + √q is irrational

Answer 2

Answer:

Suppose that p–√+q√ is a rational number equal to ab, where a and b are integers having no common factor.

Now, p–√+q√=ab

⇒p–√=ab−q√ (squaring both side)

⇒(p–√)2=(ab−q√)2

⇒p=a2b2−2(ab)q√+q

⇒2(ab)q√=a2b2+q−p

⇒2abq√=a2+b2(q−p)b2

⇒q√=a2+b2(q−p)2ab

⇒q√ is a rational number. (because sum of two rational numbers is always rational)

This is a contradiction as q√ is an irrational number.

Hence, p–√+q√ is an irrational number.