Step by step
The question
Question: rationalise 10/(4 − √6).
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 4 − √6 is 4 + √6.
So we multiply by 4 + √6/4 + √6.
4 + √6/4 + √6 is 1, and times 1 changes nothing.
Careful!
Times √6 alone does not work here.
4 − √6 times √6 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: 4 squared minus (√6) squared.
√6 squared is 6.
So it is 16 minus 6: 10.
No root at the bottom now.
So the bottom is 10.
3 Top: open the brackets
On top: 10 times every part of the bracket.
That is 40 + 10√6.
So we get (40 + 10√6)/10.
4 Simplify
40, 10 and 10 can all be divided by 10.
Over 1 is just the top: 4 + √6.
5 Check
Let's check with a calculator.
The start is about 6.45.
The answer is about 6.45 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.