Step by step
The question
Question: rationalise 1 over (√2 + 1).
Rationalise means: write it with no root at the bottom.
1 Find the partner
With one root at the bottom, we multiplied by that root.
With two terms, we use the partner: the same two terms, with the other sign.
The partner of √2 + 1 is √2 − 1.
So we multiply by √2 − 1 over √2 − 1.
√2 − 1 over √2 − 1 is 1, and times 1 changes nothing.
Careful!
Times √2 alone does not work here.
√2 + 1 times √2 still has a root in it.
Top times top, bottom times bottom.
2 Bottom: difference of squares
Why the partner?
a plus b, times a minus b, is a squared minus b squared.
A root squared has no root.
On the bottom: (√2) squared minus 1 squared.
√2 squared is 2.
So it is 2 minus 1: 1.
No root at the bottom now.
So the bottom is 1.
3 Top: open the brackets
On top: 1 times every part of the bracket.
That is √2 − 1.
So we get (√2 − 1)/1.
4 Simplify
Over 1 is just the top: √2 − 1.
5 Check
Let's check with a calculator.
The start is about 0.41.
The answer is about 0.41 too.
So it is right!
Let's look back.
First: find the partner.
Then: bottom: difference of squares.
Then: top: open the brackets.
Then: simplify.
Last: check.