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Square root calculator

--- --- Methodology

The square root of a number is the value that gives the number when multiplied by itself. This square root calculator returns the decimal value, simplifies the radical when it can (√72 = 6√2), says whether a whole number is a perfect square, and also squares a number or takes a cube root or any n-th root, with the working shown.

Whole numbers are simplified into the form a√b; decimals get a decimal answer.

Square root of 72 6√2
Square root, simplified6√2
Decimal value8.4852813742
Perfect square?no
Check8.48528137² ≈ 72
Show the working, step by step

  1. Write 72 as a product of primes.

    72 = 2³ × 3²

  2. Take each prime out of the radical 2 at a time: every 2 copies of a prime become one copy outside. The leftover copies stay inside.

    √72 = √(2² × 2 × 3²) = 6 × √2

  3. Multiply what is outside.

    √72 = 6√2

  4. Check the size: 72 lies between the perfect squares 64 and 81.

    8 < √72 < 9, and √72 ≈ 8.4852813742

Decimals are rounded to 10 significant figures. Simplified radicals are exact.

How to simplify a square root

A radical √n is in simplest form when no square factor greater than 1 divides n. The reliable method is prime factorisation, the same one used by the prime factorization calculator.

  1. Write n as a product of primes.
  2. Group equal primes in pairs. Each pair becomes one prime outside the root.
  3. Leftover unpaired primes stay inside. Multiply what is outside, and multiply what is inside.

√(a² × b) = a × √b for a ≥ 0, b ≥ 0

Worked example: √72

  1. 72 = 2³ × 3² = 2² × 2 × 3².
  2. The squared factors 2² and 3² come out as 2 and 3, and one 2 stays inside.
  3. √72 = 2 × 3 × √2 = 6√2 ≈ 8.4852813742.
  4. Size check: 72 lies between 64 and 81, so √72 is between 8 and 9. ✓

More examples

  • √200: 200 = 2³ × 5², so √200 = 2 × 5 × √2 = 10√2 ≈ 14.14.
  • √144: 144 = 2⁴ × 3², every prime pairs off, so √144 = 12 and 144 is a perfect square.
  • ∛54: 54 = 2 × 3³, so ∛54 = 3∛2 ≈ 3.7798, taking primes out three at a time.

Perfect squares

A perfect square is a whole number with a whole-number square root. The first ones are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and 169. Whole numbers between two perfect squares have a square root between the two roots, which gives a quick estimate: 72 is between 8² = 64 and 9² = 81. In prime-factor form a number is a perfect square exactly when every prime has an even exponent.

Squaring and taking the square root undo each other for non-negative numbers: √(x²) = x. For a negative x the result is |x|, not x, because the square root symbol means the non-negative root: √((−3)²) = √9 = 3. That is why the equation x² = 9 has two solutions, 3 and −3, even though √9 is just 3. The quadratic equation calculator handles the ± for you.

Common mistakes

  • Splitting a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7. Roots split over multiplication, √(ab) = √a × √b, but not over addition.
  • Stopping before the root is fully simplified. √48 = 2√12 is not finished, because 12 still holds a square factor. √48 = 4√3.
  • Forgetting the negative root when solving x² = n. The symbol √n alone is the positive root; the equation has both ±√n.
  • Trying a real root of a negative number with an even index. √(−4) is the imaginary number 2i.

Square roots appear most often in the Pythagorean theorem, where c = √(a² + b²), and in the quadratic formula.

Square root calculator: the worked example on this page, with its result and chart
Square root calculator: the worked example above, at a glance.

Common questions

How do I simplify a square root?

Factor the number into primes, pair up equal primes, and move one prime out of the root for every pair. For 72 = 2³ × 3², the pair of 3s and one pair of 2s come out, leaving a single 2 inside: √72 = 2 × 3 × √2 = 6√2. A root is fully simplified when no prime appears twice under it.

How do I check whether a number is a perfect square?

A perfect square is the square of a whole number: 1, 4, 9, 16, 25, … Take its square root. If the result is a whole number, as √144 = 12, the number is a perfect square. If it is not, as √200 ≈ 14.14, it is not. The calculator reports this for each whole number you enter.

How do I square a number?

Multiply it by itself: 1.5² = 1.5 × 1.5 = 2.25. Choose “The square of a number” in the calculator. The square of a negative number is positive, because a negative times a negative is positive, so (−1.5)² is also 2.25.

Can you take the square root of a negative number?

Not in the real numbers, because no real number multiplied by itself is negative. The answer exists as an imaginary number: √(−9) = 3i, where i is defined by i² = −1. The calculator explains this instead of returning an error. Odd roots such as the cube root of −8 are real and equal −2.

How do I find a cube root or a fourth root?

Choose “n-th root” and enter the index n. The same prime method applies, but primes leave the root n at a time: ∛54 = ∛(2 × 3³) = 3∛2 ≈ 3.7798. For a decimal answer without simplifying, the n-th root of x is x raised to 1/n.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.