Square Root Simplifier

Enter the number under √. The exact simplest radical and decimal value update immediately; factorization and learning steps stay tucked away until you want them.

√ Radicand

Number under the square root

Whole numbers from −1 trillion to 1 trillion. Commas and spaces are accepted. Negative values return the exact imaginary form instead of hiding the boundary.

Exact integer simplification · calculated locally

Simplest radical form

6√2

≈ 8.485281374239

Exact radical form plus decimal approximation.

36Largest square factor

6Outside √

2Left inside √

NoPerfect square?

√72 = √(36 × 2) = 6√2

Prime factorization: 2³ × 3²

Show simplification steps

Optional outside coefficient

Coefficient in c√n

Useful for expressions such as 3√72. Keep 1 for a plain square root. This page intentionally stays square-root focused; higher-index radicals belong in a general root calculator.

Formula and interpretation

Factor perfect-square components from the radicand: √(a²b) = a√b.

Simplification changes representation, not the numeric value.

Worked check

√72 can be simplified as √(36 × 2) = 6√2. Squaring 6√2 gives 36 × 2 = 72, which is a useful exact verification. The simplified radical and its decimal approximation are different representations of the same value.

Rule: Factor perfect-square components from the radicand: √(a²b) = a√b. Simplification changes representation, not the numeric value. For a final check, estimate the expected size of the answer before copying it, keep units consistent, and round only the displayed result rather than intermediate values. Re-enter one simple case by hand when possible; a result with the wrong sign or order of magnitude usually points to an input or reference-base mistake.

Practical guide and verification

Use the product first, then apply these tool-specific checks to verify assumptions, interpret the result, and hand it off safely without moving the primary workflow below generic content.

Exact radical and decimal answer different questions

A simplified radical such as 6√2 is exact, while its decimal display is rounded. Keep the radical for algebra and symbolic work; use the decimal only when a numerical approximation is appropriate. Rounding the decimal and then trying to reconstruct the radical can lose the exact factorization that made the simplification useful.

Verify by squaring the simplified form

After pulling out perfect-square factors, square the coefficient and multiply by the square-free radicand. The product should equal the original nonnegative radicand. This reverse check catches missed square factors and arithmetic slips without requiring a second calculator, and it works even when the remaining radical is irrational.

Negative radicands need an explicit number system

A negative integer has no real square root. If the tool displays an imaginary form, interpret it in the complex-number system using i² = −1. Do not mix a complex result into a workflow that expects real measurements, lengths, probabilities, or other quantities whose domain excludes imaginary values.

Prime factorization is evidence, not decoration

The factorization shows why a coefficient can leave the radical. Pair equal prime factors to make squares and leave unpaired factors inside. For large integers, use the factorization as an audit trail and compare the extracted square factor with the simplified coefficient before copying the final expression into homework or symbolic code.