Step by step
The question
Question: rationalise (√7 − 2) over (√7 + 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 √7 + 2 is √7 − 2.
So we multiply by √7 − 2 over √7 − 2.
√7 − 2 over √7 − 2 is 1, and times 1 changes nothing.
Careful!
Times √7 alone does not work here.
√7 + 2 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 2 squared.
√7 squared is 7.
So it is 7 minus 4: 3.
No root at the bottom now.
So the bottom is 3.
3 Top: open the brackets
On top: every part times every part.
√7 times √7 is 7.
Then the like terms go together: 11 − 4√7.
So we get (11 − 4√7)/3.
The numbers can't be cancelled, so we are done.
4 Check
Let's check with a calculator.
The start is about 0.14.
The answer is about 0.14 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.