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If you break down the number 1,2001{,}200 into its prime factors, you get 2×2×2×2×3×5×52 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5. Writing long chains of repeated numbers takes up space and makes patterns hard to spot.

The Shorthand for Repeated Factors

Index notation (also called exponent form) is a compact way to write repeated multiplication. In an expression like 242^4, the base is the number being multiplied (22), and the index or exponent (44) tells you how many times to multiply that base by itself.
How do we turn a scattered list of prime factors into clean index notation?

Converting Factor Chains

To write a prime factorization in index notation, group identical prime factors together, count how many copies you have of each, and arrange the bases in ascending order (from least to greatest).
What should you do when a prime factor appears only once in your tree?

Factors with an Exponent of One

Any number raised to the power of 11 equals itself, so 51=55^1 = 5. By convention, we omit the exponent 11 and write single factors as just the base number.
For example, 360=2×2×2×3×3×5360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5. In index notation, this becomes 23×32×52^3 \times 3^2 \times 5.
When calculating with powers, there is one critical trap that catches many students off guard.