How can prime factorization and exponentiation provide new perspectives of numbers?

Students analyze numbers using prime factorization and exponentiation.

The order in which three or more numbers are multiplied does not affect the product (associative property).

Any composite number can be expressed as a product of smaller numbers (factorization).

Prime factorization represents a number as a product of prime numbers.

Any composite factor of a number can be determined from its prime factors.

Repeated multiplication of identical factors can be represented symbolically as a power (exponentiation).

A power, A^n, includes a base, A, representing the repeated factor, and an exponent, n, indicating the number of repeated factors.

Any repeated prime factor within a prime factorization can be expressed as a power.

A product can be composed in multiple ways.

The prime factors of a number provide a picture of its divisibility.

Different representations of a product can provide new perspectives of its divisibility.

A power is divisible by its base.