surds and indeces

1. Surds
A surd is a square root which cannot be reduced to a rational number.
For example, √4 = 2 is not a surd.
However √5 is a surd.
If you use a calculator, you will see that √5 = 2.236067977 … and we will need
to round the answer correct to a few decimal places. This makes it less
accurate.
If it is left as √5, then the answer has not been rounded, which keeps it
exact.
Here are some general rules when simplifying expressions involving surds.
Indices
Indices are a useful way of more simply expressing large numbers. They
also present us with many useful properties for manipulating them using
what are called the Law of Indices.
For example:
The expression 25 is defined as follows:
We call "2" the base and "5" the index.

2. Law of indices
To manipulate expressions, we can consider using the Law of Indices.
These laws only apply to expressions with the same base, for example,
34 and 32 can be manipulated using the Law of Indices, but we cannot use
the Law of Indices to manipulate the expressions 35 and 57 as their base
differs (their bases are 3 and 5, respectively).
There are six rules
Rule 1:
Any number, except 0, whose index is 0 is always equal to 1, regardless of
the value of the base.
An Example:
Simplify 20:
Rule 2:
An Example:
Simplify 2-2:

3. Rule 3:
To multiply expressions with the same base, copy the base and add the
indices.
An Example:
Simplify : (note: 5 = 51)
Rule 4:
To divide expressions with the same base, copy the base and subtract the
indices.
An Example:
Simplify :
Rule 5:
To raise an expression to the nth index, copy the base and multiply the
indices.
An Example:
Simplify (y2)6:
Rule 6:

4. An Example:
Simplify 1252/3: