Step by step
The question
Question: rationalise 1/(3 − √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 3 − √2 is 3 + √2.
So we multiply by 3 + √2/3 + √2.
3 + √2/3 + √2 is 1, and times 1 changes nothing.
Careful!
Times √2 alone does not work here.
3 − √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: 3 squared minus (√2) squared.
√2 squared is 2.
So it is 9 minus 2: 7.
No root at the bottom now.
So the bottom is 7.
3 Top: open the brackets
On top: 1 times every part of the bracket.
That is 3 + √2.
So we get (3 + √2)/7.
The numbers can't be cancelled, so we are done.
4 Check
Let's check with a calculator.
The start is about 0.63.
The answer is about 0.63 too.
So it is right!
Let's look back.
First: find the partner.
Then: bottom: difference of squares.
Then: top: open the brackets.
Last: check.