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It is known that the square root of an integer is either an integer or irrational.

If we square root2 root3 we get 6. The square root of 6 is irrational.

Therefore, root2 root3 is irrational.

Q: How do you prove that root 2 root 3 is an irrational number?

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Yes. The square root of a positive integer can ONLY be either:* An integer (in this case, it isn't), OR * An irrational number. The proof is basically the same as the proof used in high school algebra, to prove that the square root of 2 is irrational.

(root 2)/3

That simply isn't true. The sum of two irrational numbers CAN BE rational, but it can also be irrational. As an example, the square root of 2 plus the square root of 2 is irrational.

The square root of 2 times the square root of 2 is rational.

Yes. The square of a whole number is always a whole number. For example, 3 squared is 9, so the square root of 9 is 3. What you never have, is the square root of a whole number being a fraction that is not a whole number. The square root of a whole number is either a whole number or an irrational number. For example, the square root of 2 is irrational, because there are no 2 whole numbers a and b such that a/b squared is 2. This is not terribly difficult to prove, but I have already said too much; I have answered your question.

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The square root of 2 is 1.141..... is an irrational number

This is impossible to prove, as the square root of 2 is irrational.

I linked a good resource that explains what you asked below.

No; you can prove the square root of any positive number that's not a perfect square is irrational, using a similar method to showing the square root of 2 is irrational.

Yes, the square root of 2 is an irrational number.

Yes. The square root of a positive integer can ONLY be either:* An integer (in this case, it isn't), OR * An irrational number. The proof is basically the same as the proof used in high school algebra, to prove that the square root of 2 is irrational.

The square root of 2 is an irrational number

2 is a prime number and its square root is an irrational number that cannot be expressed as a fraction

irrational

Yes. For example, the square root of 3 (an irrational number) times the square root of 2(an irrational number) gets you the square root of 6(an irrational number)

Yes, they are irrational.

It can't be expressed as a whole number. The square root of two is irrational, and the product of an irrational number and a rational number is also irrational. So 8*sqrt(2) is irrational.