prove that √p+√q is an irrational, where P, q are primes. ​

Question

prove that √p+√q is an irrational, where P, q are primes. ​

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Claire 2 months 2021-08-12T15:50:15+00:00 2 Answers 0 views 0

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    0
    2021-08-12T15:51:45+00:00

    Step-by-step explanation:

    Prime number has factors one and itself.

    So prime number cannot be a square of a number.

    Therefore,the square root of a prime number is irrational.

    If p is prime then √p is irrational.

    If q is prime then √q is irrational.

    Sum of two irrational numbers is always irrational.

    So,√p+√q is always irrational

    Hence proved

    0
    2021-08-12T15:52:07+00:00

    Answer:

    First, we’ll assume that p and q 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=x2

    p+2pq+q=x2

    2pq=x2−p−q

    pq=2(x2−p−q)

    Now, x, x2, p, q, & 2 are all rational, and rational numbers are closed under subtraction and division.

    So, 2(x2−p−q) is rational.

    But since p and q are both primes, then pq is not a perfect square and therefore pqis not rational. But this is contradiction. Original assumption must be wrong.

    So, p and q is irrational, where p and q are distinct primes.

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