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.

Example

Example

Example

Example

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

Example

aⁿ × 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}

Example

a must be non-zero.

Example

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 a^{m × n}.

Example

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.

Example

Example

Monotony of exponential function

An exponential function is a function of the form

If 0 < a < b then:
  if x > 0 => a^x < b^x
  if x < 0 => a^x > b^x
Example

Exponents - practice(problems with solutions)