Question

prove that root3 +root 5 is irrational

Answer 1

It depends a bit on what you are allowed to use. Can you use that the product of two rational numbers is rational? (If not, that's easy to prove.)

So assume that \(\Large \sqrt{3}+\sqrt{5}\) is rational.

Then it follows (since product of rationals is rational) that \(\Large (\sqrt{3}+\sqrt{5})^2\) is rational as well.

Multiply that out, and you will get a sum of a rational, and 2* a square root of something.

Now, can you use that THAT square root IS irrational? And also that the sum of a rational and irrational is irrational? If you can use all that, then you have your contradiction.

If you can't use that the square root is irrational, then you'll have to also prove that. But that should be easier to prove than that the sum of the two roots is irrational.