Question

prove 3+2underoot 5 is irrational ​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

Rational numbers are a field and it is close to addition, subtraction, multiplication and division (except division by zero).

So let's suppose that 3+25–√ is rational.

If 3+25–√ is rational, then (3+25–√)−3=25–√ is rational. And if 25–√ is rational, then 25–√÷2=5–√ would be rational.

Now, if 5–√ is rational, then there would exist to integers p,q such as 5–√=pq . If q is negative, then 5–√=−p−q for a positive integer −q . And, among all possible values of pq=5–√ with positive q there is one pair with the less possible positive q .

So let's take this value: pq=5–√ for the less possible positive integer q , which does exist if 5–√ is rational.

Then (pq)2=5 , meaning p2q2=5 and therefore p2=5q5 .

Right side is divisible by 5 . If 5 does not divide p then p2 won't be divisible by 5 . So p=5k and p2=25k2 .

Now we have 25k2=5q2 . We can simplify and we get that 5k2=q2 . Left side is divisible by 5 so q must be divisible by 5 . Let q=5h . We know that 0<h<q .

Let's replace and we get 5k2=25h2 from which we get that k2h2=5 and kh=5–√ .

With h positive and less than q . So there is no minimum positive integer q such as 5–√=pq (with p integer). So 5–√ is not rational. So 25–√ is not rational either. So 3+25–√ is irrational.

QED.

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