Prime Factorization & Exponentiation

prime factorization

• Every composite number can be broken down into a product of prime numbers. 

• Divisibility: You can tell if a number is divisible by another just by looking at its prime factors.

• Factorization: Breaking a composite number down into small numbers that multiply to reach it.

• Prime Factorization: Breaking a number down until only prime numbers are left. 

  • • Example on the right: 24 can be broken down into its factors: 2 x 12. 2 is prime, so that "branch" of the factor tree is done. 12 is composite, so it is broken down into the factors 2 x 6. 2 is prime, so that "branch" of the factor tree is done. 6 is composite, so it is broken down into the factors 2 x 3. Both 2 and 3 are prime, so those "branches" are done. The final answer is written in order from least to greatest: 24 = 2 x 2 x 2 x 3.

• Divisibility: You can determine if a number is divisible by looking at its prime factors. 

powers (exponentiation)

• Powers represent repeated multiplication. 

  • • Aⁿ, A is the Base (the factor) and n is the Exponent (how many times it repeats).

      • Eg. 2 x 2 x 2 = 2³

• You can simplify an equation from prime factorization, by replacing repeated multiplication with an exponent. 

  • • Eg. In the prime factorization of 24 (as seen above):

      • • • 2 x 2 x 2 x 3 = 2³ x 3