Question

4. Prove that root p+ root q is irrational, where p, qare primes.
(please give ans fast)​

Answer 1

Answer:

Now x, x², p, q and 2 are all rational, and rational numbers are closed under subtraction and division. So (x² - p - q) / 2 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 a contradiction

Answer 2

Answer:

First, we'll assume that √p+√q is rational, where p and q are distinct primes

√p+√q=x, where x is rational

Rational numbers areclosed under multiplication, so if we square both sides, we still get rational numbers on both sides.

(√p+√q)²=x²

p+2√(pq)+q=x²

2√(pq)=x²-p-q

√(pq)=x²-p-q)/2

Now x, x²,p, q and 2 are all rational, and rational numbers are closed under subtraction and divison. So (x²-p-q)/2 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 a contradiction. Original assumption must be wrong.

So √p +√q is irrational, where p and q are distinct primes

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We can also show that √p +√q is irrational, where p and q are not-distnict primes, i.e. p=q

we use some method: Assume √p +√q is rational.

√p+√q=x, where x is rational

√p+√q=x

2√p=x

√p=x/2

Since both x and 2 are rational, and rational numbers are closed under division, then x/2 is rational. But Since p is not a perfect square, then √p is not rational. But this is a contradiction. Original assumption must be wrong.

So √p+√q is irrational, where P and Q are non-distnict primes.

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√p+√q is irrational, where p and q are primes.

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