Exponents are just the number of times a factor is repeated. Consider 64.
64
32,2
16,2,2
8,2,2,2
4,2,2,2,2
2,2,2,2,2,2
There are six twos. That's 26
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Choose the lowest of the exponents. The GCF of x3 and x5 is x3
The GCF is 3. No exponents needed.
2 x 52
As a product of its prime factors in exponents: 22*53 = 500
If you construct them correctly, factor trees always work to determine the prime factorization of a number. Once you compare the prime factorizations of two or more numbers, it is relatively easy to find the greatest common factor of them from there.