Step by step
The question
Question: rationalise 7/(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: 7 times every part of the bracket.
That is 21 − 7√2.
So we get (21 − 7√2)/7.
4 Simplify
21, 7 and 7 can all be divided by 7.
Over 1 is just the top: 3 − √2.
5 Check
Let's check with a calculator.
The start is about 1.59.
The answer is about 1.59 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.