Step by step
The question
Question: rationalise 6 over (√3 + 1).
Rationalise means: write it with no root at the bottom.
Books always do this: a plain number at the bottom is easier to work with.
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 middle sign changed.
The partner of √3 + 1 is √3 − 1.
Why?
When we multiply them, the middle parts cancel, and the root is gone.
We see it in the next step.
So we multiply by √3 − 1 over √3 − 1.
√3 − 1 over √3 − 1 is 1, and times 1 changes nothing.
Careful!
Times √3 alone does not work here.
(√3 + 1) times √3 still has a root in it.
Top times top, bottom times bottom.
2 Bottom: one square take away another
Why the partner?
Call the first term a and the second b: here a is √3 and b is 1.
Open (a + b)(a − b): a times a, minus a times b, plus b times a, minus b times b.
The two middle parts cancel.
So it is a squared minus b squared.
A root squared has no root.
On the bottom: (√3) squared minus 1 squared.
√3 squared is 3.
So it is 3 minus 1: 2.
No root at the bottom now.
So the bottom is 2.
3 Top: open the brackets
On top: 6 times every part of the bracket.
That is 6√3 − 6.
So we get (6√3 − 6)/2.
4 Simplify
The two top parts, 6 and 6, and the bottom 2: 2 goes into 6: 2 × 3 = 6.
So 2 goes into all of them.
So we divide every part by 2.
Over 1 is just the top: 3√3 − 3.
5 Check
Let's check with a calculator.
The start is about 2.20.
The answer is about 2.20 too.
So it is right!
Let's look back.
First: find the partner.
Then: bottom: one square take away another.
Then: top: open the brackets.
Then: simplify.
Last: check.