WorkedMathGlossary

Square Root Calculator

Enter any whole number below to instantly get its decimal value and simplified radical form.

√72
8.485281
Simplified Radical Form
6√2
Perfect Square
No

√72 is irrational — its decimal form (8.485281) never terminates or repeats. The simplified radical form (6√2) is the exact value.

What Is a Square Root?

The square root of a number is the value that, when multiplied by itself, produces that number. If the number is a perfect square (1, 4, 9, 16, 25...), its square root is a whole number. Otherwise, the square root is irrational — its decimal form never terminates or repeats — and it's usually expressed in simplified radical form instead.

How to Simplify a Square Root, Step by Step

  1. Find the prime factorization of the number.
  2. Group the prime factors into pairs of identical primes.
  3. For each pair, move one copy of that prime outside the radical sign.
  4. Multiply everything left outside together, and everything left inside together.

Worked example

Simplify √72.

Prime factorization: 72 = 2^3 × 3^2

The pair of 3s comes out as a single 3. Of the three 2s, one pair comes out as a single 2, leaving one 2 behind inside the radical.

√72 = 3 × 2 × √2 = 6√2 (≈ 8.485)

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Frequently Asked Questions

What is a square root?

The square root of a number is the value that, multiplied by itself, gives that number. The square root of 9 is 3, because 3 × 3 = 9. Every positive number has two square roots (one positive, one negative), but "the square root" usually refers to the positive one, called the principal square root.

Can a square root be negative?

Technically every positive number has two square roots — a positive one and a negative one (both 3 and −3 square to 9). By convention, the √ symbol always refers to the principal (positive) square root, so √9 = 3, not −3. You only see the negative root when solving an equation like x² = 9, which has two solutions: x = 3 and x = −3.

How do you simplify a square root by hand?

Find the prime factorization of the number. For every pair of identical prime factors, move one copy outside the radical. Whatever's left without a pair stays inside. For example, 72 = 2³ × 3² — the pair of 3s comes out as 3, and one 2 pairs up leaving one 2 inside: 72 = 2² × 2 × 3² → 6√2.

What is a perfect square?

A perfect square is a number that's the square of a whole number — 1, 4, 9, 16, 25, 36... Its square root is always a whole number with no radical needed.

Is the square root of a prime number always irrational?

Yes. A prime number's only factors are 1 and itself, so it can never be split into a pair of identical factors — its square root can't simplify to a whole number or a smaller radical, and it never terminates or repeats as a decimal.

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