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Prove that √p+√q is irrational, where p, q are primes. |
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Answer» Answer: HOPE it helps PLEASE MARK ME AS BRAINAILIST PLEASE PLEASE PLEASE PLEASE PLEASE PLEASE!!!! Step-by-step explanation: First, we'll assume that p
and
is rational , where p and q are distinct primes p
+ q
=x, where x is rational Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides. ( p
+ q
) 2 =x 2
p+2 pq
+q=x 2
2 pq
=x 2 −p−q pq
= 2 (x 2 −p−q)
Now, x, x 2 , p, q, & 2 are all rational, and rational numbers are closed under subtraction and division. So, 2 (x 2 −p−q)
is rational. But since p and q are both primes, then pq is not a perfect square and therefore pq
is not rational. But this is contradiction. Original assumption must be wrong. So, p
and q
is IRRATIONAL, where p and q are distinct primes. December 20, 2019 |
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