Exponents
When you multiply a by itself, it's easy to write it down as a⋅a. However, when you have to multiply a by itself, let's say, forty five times, you need to write a very long line, which is inconvenient. You can use a shorter notation instead, called exponent (power). Remember that exponent can be used only when multiplying a number or a variable by itself. Then, instead of writing your number 45 times in a row, you can simply write a^{45}. The superscript, in this case 45, indicates the number of multipliers in the function, and it's called an exponent, and a is called a base.
An exponent can take both positive and negative values. When it's negative, the following rule applies:
$x^{a} = \frac{1}{x^a}$,
but to avoid dividing by zero the value of x must be nonzero.
The exponents can be both rational or irrational. When the exponent is fractional, for example $\frac34$, this means that you have to take the 4th degree root of x^{3}. For more details read materials for radicals.
Laws of Exponents:
Here are the basic properties, which you have to remember.
a^{n}⋅a^{m} =
a⋅a⋅a⋅a⋅a...(n times)⋅a⋅a⋅a⋅a⋅a.....(m times),
which is equal to a⋅a⋅a⋅a⋅a....(n + m times)
or
a^{m + n}

=  a^{nm} 
This case is opposite to the previous one, but in this case a must be nonzero.
If you have (a^{n})^{m}, this means (a⋅a⋅a⋅a⋅a...(n times))⋅(a⋅a⋅a⋅a⋅a...(n times))⋅(a⋅a⋅a⋅a⋅a...(n times)) ...... (m times).
In this case the number of multipliers is n times m
and (a^{n})^{m} equals a^{m⋅n}.
If you have (a⋅b)^{n} this is the same as (a⋅b)⋅(a⋅b)⋅(a⋅b)....(n times), which
equals
(a⋅a⋅a⋅a⋅a...(n times)) ⋅ (b⋅b⋅b⋅b⋅b....(n times)) or a^{n}⋅b^{n}.
This case is the opposite of the previous one (b must have a nonzero value).
a^{n}  = 

Monotony of the exponential function
If 0 < x < y then:
 if r > 0 => x^{r} < y^{r}
 if r < 0 => x^{r} > y^{r}
If x < y and they are both rational
if 0 ≤ a < 1 => a^{x} > a^{y}
if a > 1 => a^{x} < a^{y}