Algebra : Simplification
Simplification of Expressions
By simplification, I mean reduction of expressions like
to the expression
8x
Obviously it gets harder than this
Stating the above a little more formally
The first thing to realise is that the above is NOT an equation  it is an identity, i.e. it is valid for all values of x, or for whatever x represents. If x is apples then
and more conventionally (as far as Mathematics is concerned), if x represents a number like 2 (for example) then
which is obviously true.
The reason I stressing this point particularly is that such substitution can be used to test any simplification you make, if you are uncertain about what you have done.
Avoid the tendency to incorrectly 'oversimplify'. By this I mean the tendency to try and reduce everything to a single term.
for example
cannot be simplified any further. To use the analogy above, it is like having an expression like
Any attempt to simplify further will be wrong.
Rationalising Denominators
Consider
The problem with this expression is that we normally like the denominator to be an integer. However surds in the numerator are acceptable, if there is no alternative.
In order to transform the above expression into the desired form, we can multiplt top and bottom by the same number, in this case the number in the denominator. We are using two rules here

The fact that a fraction can be manipulated at will, and it will still represent the same number, as long as you either multiply top and bottom by the same number, or divide top and bottom by the same number.
A square root multiplied by a square root will, by definition, give you the number which was being squared rooted originally, e.g.
So, for our original example, multiply top and bottom by the denominator, i.e. to produce.
This is the same number as before, but in a 'nicer' form.