If the exponent is increased by 1 the value of the expression is multiplied by the base. For example, 5 to the third power is 5 X 5 X 5 = 125 and if the exponent is increased by 1 the value of the expression is 5 X 5 X 5 X 5 = 625. Increasing the exponent by 1 multiplied the value of the original result by 5.
If the exponent is decreased by 1 the value of the expression is divided by the base. For example, 5 to the third power is 5 X 5 X 5 = 125 and if the exponent is decreased by 1 the value of the expression is 5 X 5 = 25. Decreasing the exponent by 1 divided the value of the expression by 5.
Doubling the exponent squares the value of the expression. For example, 5 to the third power is 5 X 5 X 5 = 125 and if the exponent is doubled, 5 to the sixth power is 5 X 5 X 5 X 5 X 5 X 5 = 15,625 and 15,625 = 125 X 125.
Halving the exponent is equivalent to taking the square root of the value of the expression. For example, 5 to the fourth power is 5 X 5 X 5 X 5 = 625 and if the exponent is halved, 5 to the second power is 5 X 5 = 25 and 25 is the square root of 625.
To generalize the rules for activity on exponents use the concept of "precedence of operators." There are chains of operators that seem similar but each operation in the chain is more "complex" than the "simpler" operation that proceeded it. One of these chains is addition--multiplication--exponentiation. Another such chain is subtraction--division--roots.
An operation on the base-exponent expression is equivalent to a simpler operation on the exponent only. Addition on exponents is equivalent to multiplication on the base-exponent expression (addition is simpler than multiplication). Subtraction on exponents is equivalent to division on the base-exponent expression (subtraction is simpler than division). Multiplication on exponents is equivalent to exponentiation on the base-exponent expressions (multiplication is simpler than exponentiation). Division on exponents is equivalent to taking roots on the base-exponent expressions (division is simpler than taking roots).
There are no easily generalizable rules for activities on bases. You might be able to find a relationship between 5 X 5 X 5 = 125 and 6 X 6 X 6 = 216 but the relationship will not generalize to other bases.