Exponents
Exponents are algebraic operators that are used to multiply a number by itself a certain number of times. Exponents are also known as powers of numbers. Exponents are written as
where a is referred to as the base and n is known as the exponent, and the whole expression is said to be: a raised to the power of n.
is the same as saying: multiply a by itself n times. i.e.
where the last term is the nth term.
An exponent can be positive or negative, whole or fractional, a value or a variable and all these cases are dealt with differently.
Positive Exponents
Positive Exponents or powers have the effect of multiplying the base by itself as many times as the value of the exponent. As a general form, we say that if n is positive, then
is the same as multiplying a by a, n times.
For example
is the same as multiplying 3 by itself 4 times, i.e.
Raising a number or variable by the power of two is in effect squaring the number.
Similarly, raising a number or variable to the power of three is in effect cubing the number.
Negative Exponents
Negative exponents or powers have the effect of reciprocating the number or variable on which they appear, i.e.
observe that the denominator now contains the base raised to the positive power. In other words, numbers raised to negative exponents are the same as finding the reciprocal or inverse of the same number raised to the positive base. (Reciprocal means 1 divided by the number).
For example
The only exception to the rule above is if the base is zero. Remember that dividing by zero is not allowed, and zero raised to a negative power would result in 1 divide by zero:
which is not allowed.
Special Exponents
Certain numbers as exponents cause the base to behave in a special way.
Zero Exponent
As a general rule, any variable or nonzero number raised to the power zero is equal to one.
This is always true regardless of what the base a is. The base can be a nonzero number or a variable, positive or negative and, as long as it is raised to the power zero, it will equal to one. The only exception is the number zero itself because when zero is raised to the power zero, the result is still zero.
So it is important to remember that any other number, except zero, when raised to the power of zero is equal to one. For example:
Positive One Exponent
The number one is another interesting exponent. Any number raised to the power of positive one is equal to the number itself. This is regardless of whether the number is positive or negative.
and
Odd and Even Whole Number Exponents
When a positive number is raised to an odd or even whole number, the result is always positive. Negative numbers, on the other hand, behave differently.
When a negative number is raised to an even whole number power, the result will ALWAYS be a positive number. For example:
The above is also true if the exponent is a negative whole number:
You can try out different negative numbers and raise them to even whole numbers and the result is always a positive number.
Conversely, when a negative base is raised to an odd whole number exponent, the result is ALWAYS a negative number. For example:
the same applies to negative exponents:
Exponents of Ten
Ten (10) is a special number because raising ten to any whole number exponent is the in effect adding a number of trailing zeros to 10, and these are as many as the value of the exponent, i.e.
where the last 0 is the nth zero.
For example:
Fraction Exponents
Exponents can be whole numbers or fractions. Whole number exponents have been discussed above. Fractional Exponents behave different, instead of having the effect of multiplication of the base by itself, they have the effect of finding the root of the base. i.e.
is the same as finding the nth root of a, which is also written as:
Thus to find the square root of a number, we can also write:
n can be positive or negative, but in order to get real roots, a must be positive. Negative a would result in complex roots. For example;
Properties of Exponents
Numbers or variables with exponents satisfy the following properties which makes them easy to manipulate.

If the bases in a given exponential equation are equal, the exponents are also equal.
i.e.
would imply that
More on this later.

When multiplying two numbers or variables with the same base, the effect is the same as adding their exponents.
i.e.
The above is true regardless of what the base is or what values the exponents have. For example:
This can be proved as shown below:

When dividing two numbers or variables with the same base, the effect is the same as subtracting their exponents from each other as shown below:
Observe that the exponent of the denominator is subtracted from the exponent of the numerator.
For Example:
Which we can prove as follows:

Exponents distribute equally into parentheses as shown below

Exponents can only be multiplied or divided if they are in the following form:

Exponential operations are NOT associative i.e.
When solving expressions such as the one above, follow the procedure as in the example below:
First only consider the exponent of 4
then substitute the answer back
Solving Exponential Equations
Exponential equations are those with polynomials as exponents, for example:
Ordinarily, such exponential equations would be solved by using logarithms (refer to section on logarithms), but some exponential equations can be solved using the property mentioned above
applying this property to the equations above:
would become
which then implies that x = 2 since the bases are the same
Similarly,
is the same as
from which you would proceed to solve for x.
From the above example, you should have noticed that we had to put all the components of the exponential equation into the same base in order to apply that property.
So this property will only apply if the bases can be expressed in one base. For example, while the property can be used to solve the equation below
because 128 and 16 can be expressed as exponential functions of the same base, i.e.
which using another of the properties mentioned before becomes
which then leads to:
from which you can solve for x (refer to section on polynomials)
The same can not be said of the exponential equation below:
This is because 2 and 81 can't be expressed into exponential equations of the same base. This kind of exponential equation can however be solved using logarithms.
A sure way to tell if the bases can be expressed into exponential equations of the same base is to check if they are multiples of each other. In other words, check if they share a common factor that factors both of them completely leaving no remainder.
In the examples above, 128 and 16 are multiples of each other and share 2 as a common factor. In the second example, 2 and 81 are not multiples of each other and share not factors.
For a more in depth explanation on exponential functions, see exponential functions in Precalculus.