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


a1 = a


a-n =
1
an


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


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
am
= an-m

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.

(an)m = am × n


(an× bm)p = anp×bmp

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 × bn = (a × b)n


$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

Monotony of exponential function

An exponential function is a function of the form

f(x) = ax

If 0 < a < b then:
  if x > 0 => ax < bx
  if x < 0 => ax > bx

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