Simplify Radicals Calculator — Square Root in Simplest Form
Enter a non-negative integer to get its square root in simplest radical form — the calculator pulls out the largest perfect-square factor and writes √N as k√m.
- 1
Upper search bound ⌊√N⌋
⌊√72⌋ = 8Scan downward from here to find the largest k with k² dividing N. - 2
Largest perfect-square factor k²
k = 6, k² = 6² = 36 - 3
Remaining radicand m = N ÷ k²
72 ÷ 36 = 2 - 4
Decimal ≈ k × √m
6 × √2 = 8.485281
How does this calculator work?
√N = k√m where k is the largest integer with k² dividing N and m = N/k² is square-free. Scan from ⌊√N⌋ downward for the first i with N mod i² = 0. If k = 1 the radical is already simplified. Example: √72 → k = 6 (since 36 | 72) → 6√2.
Formula
How this is calculated
To simplify √N, find the largest integer k whose square k² divides N exactly. The remaining radicand is m = N ÷ k². By the product rule √(k²m) = k√m, so the simplified form is k√m. When m = 1, N is a perfect square and the result is the integer k. When k = 1, N has no perfect-square factor greater than 1 and is already in simplest form.
The algorithm scans from i = ⌊√N⌋ downward and stops at the first i where N mod i² = 0. Starting from the largest possible value guarantees that the first match gives the largest perfect-square factor — no smaller scan value could yield a larger extraction. The scan requires at most ⌊√N⌋ steps.
Only non-negative integers are accepted because the square root of a negative number is not a real number. For decimal radicands, multiply by a power of 10 to clear the decimal before simplifying, then divide by the same power afterward.
Frequently asked questions
A square root √N is in simplest form when the radicand N contains no perfect-square factor other than 1. For example √12 is not simplified because 4 = 2² divides 12; the simplest form is 2√3 (since 12 = 4 × 3).
Radical simplification in the classic sense (extracting integer square roots) is defined for integer radicands. For a decimal like √4.5, you can rewrite it as √(9/2) = 3/√2 = 3√2/2 — but that requires a separate rationalization step beyond the scope of this tool.
Simplifying gives an exact result in the form k√m, which is mathematically equivalent to √N with no rounding. The decimal approximation shown is a floating-point estimate. When exact answers matter (e.g. in proofs or geometry), the simplified radical form is preferred.
Also known as
TG we-Calculate Editorial Team. (2026). Simplify Radicals Calculator — Square Root in Simplest Form [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/simplify-radicals-calculator
TG we-Calculate Editorial Team. "Simplify Radicals Calculator — Square Root in Simplest Form." TG we-Calculate. 2026. https://we-calculate.com/calculator/simplify-radicals-calculator.
TG we-Calculate Editorial Team, "Simplify Radicals Calculator — Square Root in Simplest Form," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/simplify-radicals-calculator
@misc{wecalculate_simplify_radicals_calculator, title = {Simplify Radicals Calculator — Square Root in Simplest Form}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/simplify-radicals-calculator}}, year = {2026}, note = {TG we-Calculate} }
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