Algebra 1 Foundational

Simplifying Radicals

Simplify √n by pulling out perfect square factors — shows prime factorization and step-by-step simplification. Works for square roots (√) and cube roots (∛). Enter any coefficient and radicand to see the fully simplified form.

Live Calculator · Step-by-Step · Algebra 1
Square roots and cube roots
Prime factorization · Factor grouping
Input
The number in front of the radical sign (default: 1).
The number under the radical sign.
Please enter a valid positive integer for the radicand.
Radical Type
1 √ 72
Examples
Result
Enter a radicand and click Simplify to see the result.
Step-by-Step Solution
Product Rule for Radicals
√(a · b) = √a · √b

To simplify a square root, find the largest perfect square factor of the radicand, split the radical using the product rule, then simplify the perfect square.

Example: √72
72 = 36 · 2  →  √(36 · 2) = √36 · √2 = 6√2

Equivalently, use prime factorization:
72 = 2² · 2 · 3²  →  pull out one 2 and one 3  →  6√2

For cube roots: group prime factors in triples. Pull one prime out for each complete triple.
When Is a Radical Fully Simplified?

A radical expression is fully simplified when:

  • The radicand has no perfect square factors (for √) other than 1
  • The radicand has no perfect cube factors (for ∛) other than 1
  • There are no fractions under the radical
  • There are no radicals in the denominator
Example: √8 = 2√2 is simplified because 2 has no perfect square factors. But √8 is NOT simplified because 8 = 4 · 2 and 4 is a perfect square.
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