# 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 a45. 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 non-zero.

The exponents can be both rational or irrational. When the exponent is fractional, for example $\frac34$, this means that you have to take the 4-th degree root of x3. For more details read materials for radicals.

#### Laws of Exponents:

Here are the basic properties, which you have to remember.

an = a⋅a⋅a⋅a... (n factors of a)
a0 = 1

an⋅am = 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 am + n

am⋅an = am + n
an⋅bn = (a⋅b)n
 an am
= an-m

This case is opposite to the previous one, but in this case a must be non-zero.

If you have (an)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 (an)m equals am⋅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 an⋅bn.

(an)m = am⋅n
$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

This case is the opposite of the previous one (b must have a non-zero value).

a-n =
 1 an
$a^{\frac{1}{n}}=\sqrt[n]{a}$

#### Monotony of the exponential function

If 0 < x < y then:
- if r > 0 => xr < yr
- if r < 0 => xr > yr

If x < y and they are both rational
-if 0 ≤ a < 1 => ax > ay
-if a > 1 => ax < ay

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