Square Root Calculator
Calculate square roots with simplified radical form, perfect square check, and step-by-step simplification.
Square root
Updates as you typeMode
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Input
Number (x) ?
—
Root degree (n) ?
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Decimal precision ?
Formula
=
x1/2
Square root: √x = x1/2
Simplification: √(a² × b) = a√b
Product rule: √(a × b) = √a × √b (a, b ≥ 0)
Quotient rule: √(a ÷ b) = √a ÷ √b (b > 0)
Imaginary: √(−x) = i√x where i = √−1
Nth root: n√x = x1/n
- √x
- The non-negative number that when squared gives x.
- Perfect square
- A number whose square root is an integer (1, 4, 9, 16, 25, …).
- Simplified radical
- Factor out the largest perfect square from under the radical sign, e.g. √72 = √(36·2) = 6√2.
- Imaginary unit (i)
- i = √−1. Square roots of negative numbers are real multiples of i.
Worked example — your number
- Radicand: —
- Largest perfect-square factor: —
- Factor: —
- Pull out the square: —
- Exact: —
- Decimal: —
Every positive real number has two square roots (positive and negative); by convention √x refers to the principal (non-negative) root. Negative inputs produce an imaginary result. The nth-root generalisation behaves differently for even and odd n.