What is the square root of a negative number?

The square root of a negative number is imaginary, denoted with i where i = √(-1). So √(-9) = √(9 × -1) = √9 × √(-1) = 3i. Imaginary numbers are used extensively in engineering and physics.

How is a cube root different from a square root?

A square root asks "what times itself equals this?" A cube root asks "what times itself three times equals this?" ³√27 = 3 because 3×3×3 = 27. Unlike square roots, cube roots of negative numbers are real: ³√(-8) = -2.

What numbers have whole number square roots?

Perfect squares have whole number roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... These are squares of integers: 1², 2², 3², etc. Most numbers have irrational square roots that go on forever without repeating.

How do I calculate square root by hand?

Use estimation and refinement. For √50: 7² = 49 (too small), 8² = 64 (too big), so √50 is between 7 and 8. Try 7.1: 7.1² = 50.41 (close!). Refine to 7.07² = 49.98. Continue until desired precision is reached.

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Square Root Calculator

Find square roots, cube roots, and nth roots

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Last verified: February 2026Accuracy tested

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Root Type

Number

√72

0.000000

Simplified Radical Form

6√2

Prime Factorization

2^3 × 3^2

Squared

72.0000

Cubed

610.9403

Perfect Squares (1-225)

Root Properties

Product Rule

√(a×b) = √a × √b

Quotient Rule

√(a/b) = √a / √b

Power Rule

(√x)² = x

Rational Exponent

√x = x^(1/2)

Negative Under Even Root

√(-x) = i√x

Odd Roots of Negatives

∛(-x) = -∛x

Sources & Methodology

Formula: √x = x^(1/2)

Square and nth root calculation

Source: Standard mathematical operations

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How to Use the Square Root Calculator

The square root calculator finds square roots, cube roots, and nth roots of any number. Get both the decimal answer and simplified radical form, plus learn the prime factorization method step-by-step.

What is a Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number:

√25 = 5 because 5 × 5 = 25

The symbol √ is called the radical sign.

Simplifying Square Roots

To simplify a square root, find perfect square factors:

Factor the number into prime factors
Pair up matching primes
Bring pairs outside the radical

Example: √72 = √(36 × 2) = √36 × √2 = 6√2

Perfect Squares

Numbers whose square roots are whole numbers:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...

Cube Roots and Higher

Cube roots (³√) find what number multiplied by itself three times gives the original:

³√27 = 3 because 3 × 3 × 3 = 27

The nth root (ⁿ√) extends this to any power.

Square Roots of Negative Numbers

The square root of a negative number is an imaginary number, denoted with i:

√(-1) = i, so √(-9) = 3i

Related Calculators

For powers and exponents, use our exponent calculator. For logarithms, try our log calculator.

Frequently Asked Questions

Factor the number into perfect squares times a remainder. √72 = √(36 × 2) = √36 × √2 = 6√2. Find the largest perfect square factor. Prime factorization helps: 72 = 2³ × 3² = (2 × 3)² × 2 = 36 × 2.