What are the elements of exponentiation in natural numbers and what are the properties of exponentiation in natural numbers?

The exponent determines the number of times the base is multiplied by itself.
A number raised to 0 is equal to 1.
A number raised to 1 is equal to itself.
The product of powers with the same base is another power with the same base and whose exponent is the sum of the exponents.
The division of powers with the same base is another power with the same base and whose exponent is the difference of the exponents.
The power of a power is another power with the same base and whose exponent is the product of the exponents.
The product of powers with the same exponent is another power with the same exponent and whose base is the product of the bases.
The quotient of powers with the same exponent is another power with the same exponent and whose base is the quotient of the bases.