Step by step
The question
Question: rationalise 2 over (√2 + 2).
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 + 2 is √2 − 2.
So we multiply by √2 − 2 over √2 − 2.
√2 − 2 over √2 − 2 is 1, and times 1 changes nothing.
Careful!
Times √2 alone does not work here.
√2 + 2 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 2 squared.
√2 squared is 2.
So it is 2 minus 4: −2.
No root at the bottom now.
So the bottom is −2.
3 Top: open the brackets
On top: 2 times every part of the bracket.
That is 2√2 − 4.
So we get (2√2 − 4) over −2.
4 Simplify
The bottom is negative.
We divide the top and the bottom by negative 1: every sign changes.
Now the bottom is 2.
4, 2 and 2 can all be divided by 2.
Over 1 is just the top: 2 − √2.
5 Check
Let's check with a calculator.
The start is about 0.59.
The answer is about 0.59 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.