Multiplying powers of the same base joins two strings of repeated factors.
For positive integer m and n, aᵐ has m factors of a and aⁿ has n factors. Their product has m+n factors. The array counts aᵐ groups of aⁿ objects. Other exponent rules follow by consistent extension, with restrictions such as a≠0 for negative powers.
The rule adds exponents only when multiplying powers with the same base.