Step by step
The question
Question: rationalise (1 + √5)/(3 − √5).
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 − √5 is 3 + √5.
So we multiply by 3 + √5/3 + √5.
3 + √5/3 + √5 is 1, and times 1 changes nothing.
Careful!
Times √5 alone does not work here.
3 − √5 times √5 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 (√5) squared.
√5 squared is 5.
So it is 9 minus 5: 4.
No root at the bottom now.
So the bottom is 4.
3 Top: open the brackets
On top: every part times every part.
√5 times √5 is 5.
Then the like terms go together: 8 + 4√5.
So we get (8 + 4√5)/4.
4 Simplify
8, 4 and 4 can all be divided by 4.
Over 1 is just the top: 2 + √5.
5 Check
Let's check with a calculator.
The start is about 4.24.
The answer is about 4.24 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.