Friendly Surds

The Problem:

Square roots of most numbers are irrational – infinitely long strings of non-recurring decimal digits, with no discernible pattern to them.

Sometimes, however, you find rather strange things happening…

Try adding together.

Then square this new number.

What have you accidentally found the value of?

So, , for some integer value N.

Now try finding what

give you the value of.

Can you make up your own combination of ‘friendly’ surds that behave like this?

You can use more than two surds, or subtraction instead of addition…