How to Solve Prime Factorization with Exponents?

How to solve prime factorization with exponents, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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.

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.

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\(\color{blue}{48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3}\).
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.

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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Prime factorization with exponents: common numbers

Use this table to check your answers. Each number is written as a product of primes, with exponents for repeated factors. A prime with no exponent appears once.

Where Prime Factorization With Exponents Goes Wrong

• Stopping at a composite number. 48 = 4 × 12 is not finished. Keep splitting until every factor is prime.
• Including 1. 1 is not prime, so it never appears in a prime factorization.
• Multiplying the exponent instead of counting. 2 × 2 × 2 is 2³, not 2 × 3 or 6.
• Forgetting to check. Multiply your answer back out. For 2⁴ × 3, 16 × 3 = 48, so it is correct.
• Mixing up the order. 3² × 2² is equal to 2² × 3², but most teachers expect the smallest prime first.

Frequently Asked Questions

What is the prime factorization written with exponents?

It is a way of writing a number as a product of primes, where any prime that repeats is written as a power. For example, 72 = 2 × 2 × 2 × 3 × 3, which becomes 2³ × 3². The base is the prime and the exponent tells how many times it appears. Primes are listed from smallest to largest.

How do you write prime factorization with exponents?

Use a factor tree or repeated division to break the number into primes, then group the repeats. For 54, divide by 2 to get 27, then by 3 three times: 27, 9, 3, 1. That gives 2 × 3 × 3 × 3, which is 2 × 3³. Order the factors from least to greatest, and leave an exponent of 1 unwritten.

What is the prime factorization of 100 using exponents?

100 = 2² × 5². Split 100 into 10 × 10, and each 10 is 2 × 5. That gives 2 × 2 × 5 × 5. Two 2s become 2² and two 5s become 5². Check it: 4 × 25 = 100. Because both exponents are even, 100 is a perfect square, and its square root is 2 × 5 = 10.

What is the prime factorization of 75 with exponents?

75 = 3 × 5². Since 75 is odd, start with 3: 75 ÷ 3 = 25. Then 25 = 5 × 5. So 75 = 3 × 5 × 5, and the two 5s become 5². Check it: 3 × 25 = 75. The 3 appears only once, so it does not need an exponent.

What is the prime factorization of 84 with exponents?

84 = 2² × 3 × 7. Divide by 2 twice: 84 ÷ 2 = 42 and 42 ÷ 2 = 21. Then 21 = 3 × 7. That gives 2 × 2 × 3 × 7, so the two 2s become 2². Check it: 4 × 3 × 7 = 84. Listing the primes in order from least to greatest gives the standard answer.

What is the prime factorization of 180 using exponents?

180 = 2² × 3² × 5. Start with 180 = 18 × 10. Then 18 = 2 × 3 × 3 and 10 = 2 × 5. Together that is 2 × 2 × 3 × 3 × 5. Two 2s become 2² and two 3s become 3², while the 5 stays as it is. Check it: 4 × 9 × 5 = 180.

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