Square Root Calculator
Get the square root as a decimal AND in simplest radical form — the answer math teachers actually want: √72 becomes 6√2, with the perfect-square factorization shown. Perfect squares are flagged as exact.
Simplifying square roots
The square root of n is the number that multiplied by itself gives n. For integers that aren't perfect squares, the exact answer is a radical: pull the largest perfect-square factor out — √72 = √(36 × 2) = 6√2. The decimal (8.485…) is irrational and never exact.
Mental anchors: perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144… A root always sits between the roots of its perfect-square neighbors, which is how you estimate √50 ≈ 7.07 (between 7 and 8, nearer 7).
Worked examples
Example 1: simplify √72
Write √72 in simplest radical form.
- Find the largest perfect-square factor of 72.72 = 36 × 2
- Pull the square out.√72 = 6√2
6√2 ≈ 8.485.
Example 2: √169 is exact
Is 169 a perfect square?
- Estimate from the neighbors: 12² = 144 and 13² = 169.144 < 169 ≤ 169
- Confirm the candidate.13 × 13 = 169 ✓
Yes — √169 = 13 exactly.
Example 3: diagonal of a square room
A 12 × 12 ft room — how long is the diagonal?
- Pythagorean theorem.√(12² + 12²) = √288
- Simplify.√288 = √(144 × 2) = 12√2 ≈ 16.97 ft
12√2 ≈ 17 feet — the √2 shortcut works for every square's diagonal.
FAQ
Why do teachers want radical form instead of the decimal?+
6√2 is exact; 8.4853 is rounded. Radical form also keeps later algebra clean — squaring 6√2 gives back exactly 72, squaring the decimal doesn't.
What's the square root of a negative number?+
Not a real number — no real number squares to a negative. In the complex numbers, √−n = i√n; this calculator stays real and says so instead of guessing intent.
Does a square root have two answers?+
x² = 25 has two solutions, +5 and −5, but the √ symbol by convention means the principal (non-negative) root: √25 = 5. The ± appears when you solve equations, not in the symbol.
How were roots computed before calculators?+
By iteration: guess g, average it with n/g, repeat — Heron's method, two thousand years old and still what CPUs refine internally. Two rounds from a decent guess gives 4+ correct digits.
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