How to Solve Prime Factorization with Exponents?

Prime factorization breaks any composite number down into a product of prime numbers. When a prime factor appears more than once, you can write it using an exponent — making the result compact and easy to work with. This skill is used in finding GCF, LCM, and in simplifying fractions and radicals.

What Is Prime Factorization?

The prime factorization of a number is the expression of that number as a product of prime numbers only. A prime number has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, 13 …). A composite number has more than two factors and can be broken down into primes.

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Example: \(\color{blue}{36 = 2 \times 2 \times 3 \times 3 = 2^{2} \times 3^{2}}\)

How to Find Prime Factorization with Exponents

Method 1 — Factor Tree

Start with the number. Find any two factors. Keep breaking non-prime factors down until all branches end in prime numbers. Write the primes using exponents.

Method 2 — Repeated Division

Divide the number by the smallest prime (2) as many times as possible. Then try 3, then 5, and so on, until the quotient is 1. Write the resulting primes using exponents.

Step-by-step example: 72

72 ÷ \(\color{blue}{2 = 36}\); 36 ÷ \(\color{blue}{2 = 18}\); 18 ÷ \(\color{blue}{2 = 9}\); 9 ÷ \(\color{blue}{3 = 3}\); 3 ÷ \(\color{blue}{3 = 1}\).

Primes: \(\color{blue}{2 \times 2 \times 2 \times 3 \times 3}\) = \(\color{blue}{2^{3} \times 3^{2}}\). Check: \(\color{blue}{8 \times 9 = 72}\) ✓

Step-by-Step Summary

1. Divide the number by the smallest prime that divides it evenly.
2. Continue dividing each quotient by the smallest available prime.
3. Stop when the quotient reaches 1.
4. Collect all prime divisors; group identical ones using exponents.
5. Check by multiplying all prime factors together to get the original number.

Watch: Prime Factorization (Video Lesson)

Math Antics walks through how to break numbers into their prime factors step by step:

Worked Examples

Example 1: Find the prime factorization of 36.

\(\color{blue}{36 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3}\).
With exponents: \(\color{blue}{2^{2} \times 3^{2}}\). Check: \(\color{blue}{4 \times 9 = 36}\) ✓
Answer: 2² × 3²

Example 2: Find the prime factorization of 48.

\(\color{blue}{48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3}\).
With exponents: \(\color{blue}{2^{4} \times 3}\). Check: \(\color{blue}{16 \times 3 = 48}\) ✓
Answer: 2⁴ × 3

Example 3: Find the prime factorization of 100.

\(\color{blue}{100 = 2 \times 50 = 2 \times 2 \times 25 = 2 \times 2 \times 5 \times 5}\).
With exponents: \(\color{blue}{2^{2} \times 5^{2}}\). Check: \(\color{blue}{4 \times 25 = 100}\) ✓
Answer: 2² × 5²

Example 4: Find the prime factorization of 180.

\(\color{blue}{180 = 2 \times 90 = 2 \times 2 \times 45 = 2 \times 2 \times 3 \times 15 = 2 \times 2 \times 3 \times 3 \times 5}\).
With exponents: \(\color{blue}{2^{2} \times 3^{2} \times 5}\). Check: \(\color{blue}{4 \times 9 \times 5 = 180}\) ✓
Answer: 2² × 3² × 5

More Practice: What Is an Exponent?

Math with Mr. J reviews exponent notation — the key tool for writing prime factorizations compactly:

Exercises

1. Find the prime factorization of 60, using exponents.
2. Find the prime factorization of 84, using exponents.
3. Find the prime factorization of 120, using exponents.
4. Which of these is the correct prime factorization of 50? (a) \(\color{blue}{2 \times 5}\)², (b) 2² × 5, (c) \(\color{blue}{5 \times 10}\).
5. Find the prime factorization of 144, using exponents.

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Answers

1. \(\color{blue}{2^{2} \times 3 \times 5}\)
2. \(\color{blue}{2^{2} \times 3 \times 7}\)
3. \(\color{blue}{2^{3} \times 3 \times 5}\)
4. \(\color{blue}{(a) 2 \times 5^{2}}\)
5. \(\color{blue}{2^{4} \times 3^{2}}\)

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Frequently Asked Questions

Does every number have a unique prime factorization?

Yes. The Fundamental Theorem of Arithmetic states that every integer greater than 1 has exactly one prime factorization (ignoring order of factors). This is why prime factorization is so useful in number theory.

What is the prime factorization of a prime number?

A prime number is already in its simplest form. Its only factorization is itself: e.g., the prime factorization of 7 is just 7 (or 7¹).

How is prime factorization used in real problems?

Prime factorization is used to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM), which in turn are used to add/subtract fractions, reduce fractions, and solve many GED word problems.

Related Topics

• Use Exponents to Write Multiplication Expressions
• Using Exponents to Write Powers of Ten
• Evaluate Integers Raised to Rational Exponents
• Simplifying Fractions
• Order of Operations