Prime Factorization is the process of breaking down a number into a product of prime numbers (numbers that have no factors other than 1 and themselves). A helpful way to find these prime factors is by using factor trees.

Steps for Prime Factorization

1. Start with the Number: Choose a number you want to factor.
2. Choose Factors: Think of two numbers that multiply to get that number. Write the number at the top, then draw two branches for the factors.
3. Repeat Until All Factors are Prime: For each composite (non-prime) factor, keep breaking it down into smaller factors. Stop when all branches lead to prime numbers.
4. List the Prime Factors: Once all factors are prime, list them.
5. Combine Like Factors with Exponents: Group any repeating prime factors using exponents to simplify the answer.

Example 1: Prime Factorization of 180

1. Start: Write 180.
2. First Split: Choose factors like $60$ and $3$.
3. Continue Factoring:
   • $3$ is already prime.
   • Break $60$ down into $30$ and $2$.
   • $2$ is prime.
   • Break 30 down into 1$5$ and $2$.
   • Break 15 into $5$ and $3$, which are both prime.
4. List Prime Factors: $2$, $2$, 3, $3$, and $5$.
5. Combine with Exponents: 180=2^2⋅3^2⋅5.

Example 2: Prime Factorization of 210

1. Start: Write $210$.
2. First Split: Choose factors like $2$ and 105.
3. Continue Factoring:
   • $2$ is prime.
   • Break 105 into $5$ and 21.
   • 5 is prime.
   • Break 21 into $3$ and $7$, both prime.
4. List Prime Factors: 2, 3, 5, and $7$.
5. Final Answer: 210=2⋅3⋅5⋅7