Step by step
The question
Question: rationalise 6 over (√7 − 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 √7 − 1 is √7 + 1.
So we multiply by √7 + 1 over √7 + 1.
√7 + 1 over √7 + 1 is 1, and times 1 changes nothing.
Careful!
Times √7 alone does not work here.
√7 − 1 times √7 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: (√7) squared minus 1 squared.
√7 squared is 7.
So it is 7 minus 1: 6.
No root at the bottom now.
So the bottom is 6.
3 Top: open the brackets
On top: 6 times every part of the bracket.
That is 6 + 6√7.
So we get (6 + 6√7)/6.
4 Simplify
6, 6 and 6 can all be divided by 6.
Over 1 is just the top: 1 + √7.
5 Check
Let's check with a calculator.
The start is about 3.65.
The answer is about 3.65 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.