Exponents and Exponential Functions

Working with Exponents
Recall that if n is a positive integer then aⁿ = (a)(a)...(a) [where there are n a's on the right hand side of this equality]. The expression a is called the base, and the n is the exponent. We often call an exponentiated expression like 2^k a power of 2, or we read 2^k as "2 to the power of k". In any exponentiated expression, you should interpret the exponent as belonging to the quantity immediately to it's left. So, for example, 2x⁵ means 2(x⁵), not (2x)⁵.
Note: any base raised to the power of 1, is just the base (i.e. a¹ = a). Also, any base raised to the power of 0 is 1 (i.e. a⁰ = 1), while 0⁰ is undefined.

Properties of Exponents:
The following hold for any base greater than 0, and any real numbers m and n.
1) a^m+n = a^maⁿ
2) a^m-n = a^m/aⁿ, a 0
3) a^mn = (a^m)ⁿ
4) aⁿbⁿ = (ab)ⁿ
5) a^m/b^m = (a/b)^m, b 0

Negative integer exponents:
A negative integer exponent is defined as: a^-n = 1/(aⁿ), where n is a positive integer and a is a real number, a 0.
For example, 5^-3 = 1/5³ or y^-1 = 1/y.

Rational Exponents
Recall from the Square Root and Other Radicals section, a^m/n = where m and n are positive integers, not both even.
Rational exponents may also be negative, and are defined in the same basic way as negative integer exponents:

                       1     
             a-m/n =  -----
                      am/n

where a 0, and m and n are positive integers (not both even).

Here are some examples.

Exponential Functions:
The function f(x) = a^x, where x is any real number and a>0 is a constant, a 1, is called the exponential function to the base a.
NOTE: The positions of the constant and the variable in an exponential function are important! The base is a constant, and the exponent is a variable (if the base is a variable, then you have a polynomial function)!!

A Special Base "e":
"e" is an irrational number like or sqrt(5). That is, the decimal expansion for e is non-repeating. The value of e is 2.718281828459045.... The symbol e will always represent this number from now on, and should not be thought of as a variable. Click here to see an example!

Algebra with e:
Algebraic manupulations with e are very important in university-level math. Click here to see some examples!

---