Exponents are a shorthand method of writing repeated multiplication of a number. That is basically all that it is.
Say that you were solving a problem and the answer turned out to be:
2*2*2*2*2*2*2*5
Writing all of the "2 times" gets rather tedious and repititious. Therefore, a more compact, easier to use method has been developed to represent this.
27*5 or 5*27
And this means 2 multiplied by itself seven times. And, if you will keep in mind that the purpose of exponents is to make life simpler then they get much easier to deal with. Specifically, keep in mind that
2*2*2*2*2*2 can be rewritten 26 and that either one of
these means 2 multiplied by itself six times.
So when you multiply 2 by itself 6 times, what do you get?
Well, one multiplication at a time you get:
2 4 8 16 32 64
So, 2*2*2*2*2*2 and 26 = 64
Of course to "say" the first of these you would say "2 times 2 times 2 times 2 times 2 times 2".
To "say" the second of these you merely say "2 to the sixth power". They both mean the same exact thing.
And if you keep the above simple example in mind, exponents are easy.
So, what is 26 * 23 ?
Well, keeping the above in mind it is
2*2*2*2*2*2 * 2*2*2
I separated these for your convenience. But what is it really in exponent notation?
Well, its 29
Just count all those "2s" above.
Clearly, it is 2 multiplied by itself nine times, which is what 29 represents.
But one more thing is obvious, if the base number is the same (2 in this case)
then when you do the math like this:
26 * 23
all you have to do is KEEP the base (2) and add the exponents (6+3).
So, the answer is 29
So, that gives you some insight into how you would multiply exponents with the same base
(just add the exponents).
But how about dividing? Well, consider
26 / 23
which means
2*2*2*2*2*2 / 2*2*2
It should be rather obvious that several 2s will cancel which will leave
2*2*2 or 23
So, a simple rule of thumb is when multiplying exponents with the same base
just add the exponents.
And, when dividing exponents with the same base just subtract the exponent of the
denominator (bottom) from the exponent of the numerator (top) and keep the base.
There clearly can be more complicated exponents than these discussed above,
but this simple rule of thumb will provide a sound guide to totally
solving many of them.
Other references:
Basic Exponent Properties Video from Patrick - Just Math Tutor