Prime factorization with exponents

key notes:

Prime Factorization:

Prime factorization is the process of breaking down a composite number into its prime factors, which are prime numbers that multiply together to give the original number.

Example: The prime factorization of 18 is 2×3×3, where 2 and 3 are prime numbers.

Exponents:

An exponent indicates how many times a number (called the base) is multiplied by itself.

Example: 3^2 (read as “three squared”) means 3×3, which equals 9.

Using Exponents in Prime Factorization:

When a prime factor repeats in the prime factorization of a number, it can be expressed using an exponent.

Example: The prime factorization of 18 can be written as 2×3^2, where the exponent 2 shows that 3 is used twice as a factor.

Steps to Find Prime Factorization with Exponents:

Step 1: Perform prime factorization by dividing the number by the smallest prime number.

Step 2: Continue dividing until you reach a quotient of 1.

Step 3: Write the prime factors, grouping identical prime numbers together.

Step 4: Use exponents to express repeated prime factors.

Example: The prime factorization of 72 is 2×2×2×3×3, which can 2^3 x 3^2

Why Use Exponents?

Exponents simplify the expression of prime factorization, making it easier to read and understand.

Example: Instead of writing 2×2×2×5 for 40, we can write 2^3×5.

Examples for Practice:

  1. Example 1: Prime Factorization of 36 with Exponents
    • Step 1: Divide by 2: 36÷2=18
    • Step 2: Divide by 2: 18÷2=9
    • Step 3: Divide by 3: 9÷3=3
    • Step 4: Divide by 3: 3÷3=1
    • Prime factorization: 2×2×3×3=2^2 x 3^2
  2. Example 2: Prime Factorization of 48 with Exponents
    • Step 1: Divide by 2: 48÷2=24
    • Step 2: Divide by 2: 24÷2=12
    • Step 3: Divide by 2: 12÷2=6
    • Step 4: Divide by 2: 6÷2=3
    • 3 is a prime number.
    • Prime factorization: 2×2×2×2×3=2^4 x 3
  3. Example 3: Prime Factorization of 100 with Exponents
    • Step 1: Divide by 2: 100÷2=50
    • Step 2: Divide by 2: 50÷2=25
    • Step 3: Divide by 5: 25÷5=5
    • Step 4: Divide by 5: 5÷5=1
    • Prime factorization: 2×2×5×5== 2^2 x 5^2

Learn with an example

🔔 Write the prime factorization of 4. Use exponents when appropriate and order the factors from least to greatest (for example, 22 . 3 . 5).

Divide by prime factors until the quotient is 1.

4÷2 = 2
2÷2 = 1
The prime factorization of 4 is:
2 . 2

Rewrite the repeated factor (2) with exponent.

22

🔔 Write the prime factorization of 8. Use exponents when appropriate and order the factors from least to greatest (for example, 22 . 3 . 5 ).

Divide by prime factors until the quotient is 1.
8÷2 = 4
4÷2 = 2
2÷2 = 1

The prime factorization of 8 is:
2 . 2 . 2

Rewrite the repeated factor (2) with exponent.
23

Write the prime factorization of 8. Use exponents when appropriate and order the factors from least to greatest (for example, 22 . 3 . 5).

Divide by prime factors until the quotient is 1.

8÷2 = 4
4÷2 = 2
2÷2 = 1

The prime factorization of 8 is:

2 . 2 . 2

Rewrite the repeated factor (2) with exponent.

23

Let’s practice!🖊️