In the following figure, three situations with powers are shown:

The table on the left shows powers of 2.  You can ignore the entry for zero, if you like.  Also, don’t be put off by the notation 2^n instead of the (probably) more familiar .  The thing to focus on is that each entry in the table is twice the value above it.  As you can see, the values in the table grow very rapidly.

The table in the middle sows powers of 4.  Each value in the table is 4 times the value above it.  The table on the right has powers of 8: each entry is 8 times the value above it.

The middle and right table are shown in a particular way that has their values line up with the table on the left.  You can see that all the values in the powers-of-4 table are also found in the powers-of-2 table.  Same with the powers-of-8 values, they are all found in the powers-of-2 table as well.  Moreover, the powers-of-4 values are found at regular places in the powers-of-2 table: they form the even entries in the powers-of-2 table.  The powers-of-8 values can be found in the powers-of-2 table at every third entry.  You can probably figure out why that must be so.  If the powers-of-2 values are all twice the value of the entry above, a value two down must be twice of twice of that value, hence 4 times as much.   Three down must be twice of twice of twice that value, hence 8 times as much.

Below, I show a graph of the powers-of-4.  Note again how fast the values grow.   In the same space, powers of 2 are shown, using the red scale for the horizontal values, and red dots to mark the values.

The black dots and scale represents and the red dots and scale represent .  Since the graph for and the graph for overlap so nicely, we are going to ask ourselves what the extra red dots mean with respect to the graph for .  In other words, if we were to draw black dots where all the red dots are, what would those new black dots mean?  This is shown below:

If we were to trust the pattern of the black dots, and treated it as a graph of for both whole and “half” values of n, we have now extended our notion of what a “power” is.  Our graph shows values for and and .  Where do these values come from?  We got them from embedding into .  We get to play Wile E Coyote for a while, standing on nothing, and hopefully not falling for a while yet.

Let’s end this post with an equation for what means even if n is a value like .5 or 1.5 or 2.5, and come back to this issue later.

The equation is:   .