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Square Root Calculator – Free Instant √ Tool with Formula & Steps
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Square Root Calculator

Instantly calculate square roots, cube roots, and any nth root — with step-by-step solutions and formulas.

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Root Formulas

Mathematical definitions and relationships used in root calculations.

Square Root
√a = a^(1/2)
A number b such that b² = a. Example: √49 = 7, because 7² = 49.
Cube Root
∛a = a^(1/3)
A number b such that b³ = a. Example: ∛125 = 5, because 5³ = 125.
Nth Root (General)
ⁿ√a = a^(1/n)
A number b such that bⁿ = a. Works for any positive integer n.
Product Rule
√(a·b) = √a · √b
The square root of a product equals the product of square roots. Example: √(4·9) = 2·3 = 6.
Quotient Rule
√(a/b) = √a / √b
The square root of a quotient equals the quotient of square roots (b ≠ 0).
Negative Radicand
√(−a) = i√a
When a > 0, the square root of a negative number is imaginary. Uses the imaginary unit i.

How to Calculate

Manual estimation method using Babylonian (Heron's) algorithm — no calculator needed.

Estimating a Square Root by Hand
1

Choose an initial estimate b close to the answer.

2

Divide the original number a by your estimate to get c: c = a ÷ b

3

Average the estimate and quotient: b = (b + c) / 2

4

Repeat steps 2–3 until the desired decimal precision is reached.

Example: Find √27 to 3 decimal places Guess: b = 5.125
27 ÷ 5.125 = 5.268 → avg = (5.125 + 5.268)/2 = 5.197
27 ÷ 5.197 = 5.195 → avg = (5.197 + 5.195)/2 = 5.196
27 ÷ 5.196 = 5.196 ✓ → √27 ≈ 5.196
Estimating an Nth Root by Hand
1

Estimate an initial value b.

2

Compute c using: c = a ÷ b^(n−1)

3

Update your estimate using: b = [b × (n−1) + c] / n

4

Repeat until your answer stabilises to the desired precision.

Example: Find ⁸√15 to 3 decimal places Guess: b = 1.432
15 ÷ 1.432⁷ = 1.405 → update = (1.432×7 + 1.405)/8 = 1.388
15 ÷ 1.388⁷ = 1.403 → update = (1.403×7 + 1.388)/8 = 1.402
Converges → ⁸√15 ≈ 1.403

Perfect Squares Reference

Common values for square roots and cube roots at a glance.

Number (n) n² √n n³ ∛n

Frequently Asked Questions

Common questions about square roots and root calculations.

A square root of a number a is a value b such that b² = a. For example, the square root of 25 is 5, because 5² = 25. Every positive number has two square roots: a positive and a negative one (±5 for 25), but by convention √ denotes the positive (principal) root.
Not within the real number system. The square root of a negative number is an imaginary number. For example, √(−9) = 3i, where i is the imaginary unit defined as √(−1). This is the basis of complex number mathematics.
A square root finds a number that when multiplied by itself (n = 2) gives the original. A cube root finds a number that when multiplied by itself three times (n = 3) gives the original. Both are special cases of the general nth root formula: ⁿ√a = a^(1/n).
When a number is not a perfect square (like 2, 3, 5, 7…), its square root is irrational — a non-terminating, non-repeating decimal. For example, √2 ≈ 1.41421356… and goes on forever with no repeating pattern.
Factor out perfect square factors from under the radical. For example: √72 = √(36 × 2) = √36 × √2 = 6√2. The goal is to remove all perfect square factors from the radicand (the number under the root sign).
Yes. √0 = 0, because 0² = 0. Zero is considered a perfect square and is the only number whose square root equals itself.
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