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And, to solve an equation, I have to get the variable by itself on one side of the "equals" sign; to isolate the variable, I have to "undo" whatever has been done to the variable.In this case, the variable has been put in the exponent.
The Hydra was a one-headed monster but when it is cut off, 2 more heads grow in its place. It might not be actually easier to find the answer (without a calculator), but isn’t is neat to be able to use an exponent to write such a long multiplication problem in such a simple way?
If a hero tried to conquer it by cutting off all of its heads every day, how many heads would the Hydra have on the third day?
What this means is that we are multiplying 3 by itself 5 times. Let's think about what a negative sign means a little more.
Before we proceed, we define 3 terms: Our definition of exponentiation makes sense if the exponent is a positive integer. When we append a negative sign to a number (say 4, for example), we are basically saying go four units in the opposite direction.
Let's say we wanted to capture the notion of "the amount equal to 3, ten times." We could write this out as , but this gets burdensome quickly: if we wanted to capture the idea of "the amount equal to two hundred 3s." Thus, we define the multiplication function, usually denoted or , such that where there are 200 threes in the sum.
This process (actually an inductive definition) defines the operation of "multiplication by positive integers." We can then extend the notion of multiplication to non-integers.
Similarly, the exponentiation is defined as the repetition of multiplication.
For example, writing out can get boring fast, so we define the exponential function to express this in a much more compact form so that the preceeding example can be written as (read 3 to the 5th or 3 to the 5 power).
Similarly, if is a positive integer, we define to be . Otherwise we'd be dividing by .) How could we make sense of an expression like ?
If you don't already know the answer, this is a good exercise; I recommend puzzling over it for awhile.