Hi Janna.
The "Fundamental Theorem of Arithmetic" (a rather grand name, so you know it is important) says that every positive integer greater than one can be represented uniquely as a product of one or more prime numbers, ignoring order of the factors.
That means that if you have the number 1710, it can be represented as 2 x 3 x 3 x 5 x 19, all of those factors are prime numbers. No other prime numbers go into 1710. Since 3 shows up twice, a more common way to show this is: 2 x 3^{2} x 5 x 19
Now let's think about 1710 as represented by "a" in your expression. That means that a^{n} is the same as the product of all the same primes above  each taken to the nth power:
1710^{n} = 2^{n} x 3^{2n} x 5^{n} x 19^{n}.
Your question says that p (a prime) divides a^{n}. Remember that no other primes besides the ones we've listed go into a^{n} (it's unique). Can you see why that means that p also divides a itself? And since it is prime, then it must be equal to 2, 3, 5, or 19 (if we use the example of a=1710).
You can also see pretty clearly that if taking a number to the nth power is the same as taking each of its prime factors to the nth power, then p^{n} is one of those new factors.
Hope this helps!
Stephen La Rocque>
