Prime Factorization Calculator · Practical guide
How to Write Prime Factorization With Exponents
Count how many times each prime repeats, then write it as a power — the compact form of any factorization.
At a glance
The Rule
First factor the number completely by trial division, listing every prime including repeats. Then group identical primes: each distinct prime becomes a base, and the number of times it appeared becomes the exponent. Write the bases in ascending order, joined by multiplication signs.
Interactive tool
Prime Factorization Calculator
Enter your own values, then compare the result with the examples below.
Result
Worked Examples
| Number | Full Factor List | Exponent Form |
|---|---|---|
| 144 | 2 × 2 × 2 × 2 × 3 × 3 | 2^4 × 3^2 |
| 360 | 2 × 2 × 2 × 3 × 3 × 5 | 2^3 × 3^2 × 5 |
| 1000 | 2 × 2 × 2 × 5 × 5 × 5 | 2^3 × 5^3 |
| 97 | 97 (prime) | 97 |
Check by Expanding Back
A factorization is only right if the powers multiply back to the original. Take 2^3 × 3^2 × 5: that is 8 × 9 × 5 = 360. If your expanded product misses the target, an exponent or a prime is wrong — recount the repeats.
Why the Exponent Form Is Useful
Beyond compactness, exponents unlock shortcuts. The total number of divisors is the product of each exponent plus one — 360 has (3+1)(2+1)(1+1) = 24 factors. And the greatest common factor of two numbers comes from taking the minimum exponent of each shared prime: for 360 = 2^3 × 3^2 × 5 and 252 = 2^2 × 3^2 × 7, the GCF is 2^2 × 3^2 = 36. The least common multiple uses the maximum exponents instead.
Frequently Asked Questions
Do I need to write the exponent 1?
No — a prime that appears once is written without an exponent. Writing 5^1 is correct but redundant, so 2^3 × 3^2 × 5 is the conventional form.
Does the order of the primes matter?
Mathematically no, since multiplication is commutative. By convention the primes are written smallest to largest, which makes answers easy to compare.
Is writing 2 × 2 × 2 × 3 × 3 × 5 wrong?
It is a correct factorization, just not in exponent form. Teachers usually ask for exponents precisely because they make counting and comparing factors easier.
How do exponents help find the GCF and LCM?
For the GCF, take the smallest exponent of each prime present in both factorizations; for the LCM, take the largest exponent of every prime present in either. Multiply the results.