Last updated: April 2026 · Reviewed by Calculatormatics Editorial Team

Square Root Calculator — Simplify Radicals Step by Step

Enter any number to instantly find its square root as both a decimal approximation and an exact simplified radical. For positive integers the calculator shows the full prime factorization, groups identical prime factors into pairs, moves each pair outside the radical as a coefficient, and leaves any unpaired factors under the radical sign. This is the same process taught in algebra classes worldwide. Simplified radical form — like 6√2 instead of 8.485… — is exact, easier to manipulate in equations, and required in many exam answers. The calculator also handles negative inputs (imaginary roots) and decimal inputs (decimal approximation only). Enable "Show simplification steps" to see every stage of the working.

Step 1 — Prime factorization:
  72 = 2³ × 3²

Step 2 — Group into pairs:
  2³ → 1 pair(s) out: 2 moves outside
  2³ → remainder 2 stays under radical
  3² → 1 pair(s) out: 3 moves outside

Step 3 — Write the simplified form:
  √72 = √(36 × 2) = √36 × √2 = 6√2

What Is a Square Root?

The square root of a number x is a value y such that y² = x. In other words, squaring the square root returns the original number: (√x)² = x. Every positive number has two square roots — one positive and one negative — because both (+y)² and (−y)² equal x. By convention, the principal square root (written √x) always refers to the non-negative root.

When both roots are needed (e.g., solving x² = 9), the ± notation is used: x = ±√9 = ±3. This distinction matters in algebra: the equation x² = 81 has two solutions (x = 9 and x = −9), but √81 = 9 only (the principal root).

Perfect Squares

A perfect square is an integer whose square root is also an integer. The first 20 perfect squares are listed below. Recognising these on sight speeds up mental arithmetic and radical simplification.

How to Simplify a Square Root

Simplifying a square root means writing it in the form a√b, where b contains no perfect-square factor (b is "square-free"). The algorithm uses prime factorization:

1. Find the prime factorization of the integer under the radical.
2. Group prime factors into pairs. Each pair of identical primes represents a perfect square.
3. Move each pair outside the radical as a single factor (one instance of that prime). For example, a pair of 3s (i.e., 3²) exits the radical as a single 3.
4. Multiply all outside factors together to form the coefficient a.
5. Multiply all remaining (unpaired) primes together; this product stays under the radical as b.
6. If b = 1, the original number was a perfect square and the result is just a.

The result a√b is the simplified radical form. It is exact and equivalent to the original √n.

Worked Example: Simplify √72

√72 is the classic textbook example because 72 has a convenient factorization. Here is the full working:

Step 1 — Prime factorization:
  72 = 2 × 36 = 2 × 4 × 9 = 2³ × 3²

Step 2 — Group into pairs:
  2³ = (2²) × 2 → one pair of 2s (exits as 2), one 2 remains
  3² = (3²)   → one pair of 3s (exits as 3), nothing remains

Step 3 — Collect outside and inside factors:
  Outside: 2 × 3 = 6
  Inside:  2 (the unpaired 2)

Step 4 — Write the result:
  √72 = √(36 × 2) = √36 × √2 = 6√2

Verification: (6√2)² = 36 × 2 = 72 ✓

Decimal check: 6 × √2 = 6 × 1.4142135624 = 8.4852813742 ✓

Worked Example: √13 (No Perfect-Square Factor)

Not every number can be simplified further. Consider √13:

Prime factorization: 13 = 13¹ (13 is prime)

No pairs exist — 13 appears only once.
All factors stay under the radical.

Result: √13 cannot be simplified further.
√13 ≈ 3.6055512755 (irrational, non-terminating)

When a number under the radical is already square-free (all prime exponents are 1), the square root is irrational and the simplified form is identical to the original. In these cases the decimal approximation is the only numeric answer.

Square Roots of Fractions and Decimals

The product rule for square roots extends to fractions:

√(a/b) = √a / √b    (for a ≥ 0, b > 0)

Example: √(9/16) = √9 / √16 = 3/4 = 0.75

Example: √(1/2) = 1/√2 = √2/2 ≈ 0.7071 (rationalised denominator)

For decimal inputs like √7.5, simplification into exact radical form is not straightforward without converting to a fraction first. The calculator displays the decimal approximation directly. If you need an exact form, convert to a fraction: 7.5 = 15/2, so √7.5 = √(15/2) = √30 / 2.

Negative Numbers and Imaginary Roots

In the real number system, no real number squared produces a negative result (because both a positive and a negative number, when squared, give a positive result). Therefore, the square root of a negative number is not a real number.

In the complex number system, the imaginary unit i is defined as i = √(−1). Using this definition:

√(−n) = √n × i    for n > 0

Examples:
  √(−4)  = √4 × i  = 2i
  √(−25) = √25 × i = 5i
  √(−72) = √72 × i = 6√2 × i ≈ 8.4853i

Imaginary and complex numbers appear in electrical engineering (impedance), quantum mechanics, signal processing, and anywhere oscillating systems are modelled. In those fields, √(−1) is sometimes written as j instead of i to avoid confusion with current notation.

Frequently Asked Questions