Square Root Calculator

Find the square root of any number, see it in simplified radical form, and switch to cube roots or any other root.

Root
Square root
–
Simplified form–
Perfect power?–
Check–

What is a square root?

Squaring a number means multiplying it by itself: 6² = 6 × 6 = 36. The square root undoes that. The square root of 36 is 6, written √36 = 6. Numbers like 1, 4, 9, 16, 25 and 36 are perfect squares because their roots are whole numbers.

Simplifying square roots

When a number is not a perfect square, its root is usually written in simplified radical form. Pull out the largest perfect square factor:

  1. Write the number as a product with a perfect square: 72 = 36 × 2.
  2. Split the root: √72 = √36 × √2.
  3. Simplify the perfect square: √72 = 6√2 ≈ 8.4853.

More examples: √50 = 5√2, √48 = 4√3, √200 = 10√2. The calculator does this automatically for whole numbers.

Cube roots and other roots

The cube root asks which number multiplied by itself three times gives your number: ∛125 = 5 because 5 × 5 × 5 = 125. More generally, the nth root of x is the number that raised to the power n equals x, and it can also be written as x1/n. Choose Nth above to find any root.

The Babylonian method

A quick way to calculate a square root by hand is to guess, then average your guess with the number divided by your guess. Repeat until it stops changing. For √20:

  • Guess 4. Then (4 + 20 ÷ 4) ÷ 2 = 4.5
  • (4.5 + 20 ÷ 4.5) ÷ 2 ≈ 4.4722
  • One more round gives 4.47214, which matches √20 to five decimal places.

Where square roots are used

  • Geometry: finding the side of a square from its area, or a triangle side with the Pythagorean theorem.
  • Statistics: the standard deviation is the square root of the variance.
  • Physics and engineering: formulas for falling objects, waves and electrical circuits.

Last reviewed October 2026.

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 49 is 7 because 7 × 7 = 49. Every positive number also has a negative square root (−7), but √ means the positive one.

How do I simplify a square root?

Find the largest perfect square that divides the number, then take its root outside the radical. For √72, the largest perfect square factor is 36, so √72 = √36 × √2 = 6√2.

What is the square root of a negative number?

There is no real-number answer, because any real number times itself is zero or positive. In complex numbers, √−1 is called i, so √−9 = 3i. This calculator works with real numbers, so enter a number of 0 or more for even roots.

What is the difference between a square root and a cube root?

A square root is a number multiplied by itself twice (x × x). A cube root is multiplied three times (x × x × x). The cube root of 27 is 3. Cube roots of negative numbers exist: the cube root of −8 is −2.

Is the square root of 2 a rational number?

No. √2 ≈ 1.41421356… goes on forever without repeating, so it cannot be written as a fraction. The square root of any whole number that is not a perfect square is irrational.

How can I estimate a square root without a calculator?

Find the two perfect squares around your number. √50 is between √49 = 7 and √64 = 8, and closer to 7, so it is a little over 7 (about 7.07). The Babylonian method below makes the estimate more precise.