. 14 Prove that √p+√q is irrational, where p and q are primes.
Answers
Answered by
0
Answer:
Step-by-step explanation:
Answer
4.5/5
63
lalitc2502
Virtuoso
79 answers
6.2K people helped
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'
Similar questions