Square Root Calculator

What a square root is

The square root of a number n is the value that, multiplied by itself, gives n. Because 9 × 9 = 81, the square root of 81 is 9, written √81 = 9. Squaring and taking a square root undo each other, in the same way that multiplying and dividing do. Every positive number has two square roots, one positive and one negative. The radical sign √ always means the positive one, called the principal square root.

Numbers such as 1, 4, 9, 16 and 25 are perfect squares because their roots are whole numbers. The square root of any other whole number is irrational, an endless, non-repeating decimal. That’s why answers are often left in exact radical form, such as 3√5, instead of a rounded decimal.

How to use the square root calculator

1. Type the number into the box. Whole numbers, decimals and negative numbers all work.
2. The decimal square root appears at once, to up to 10 decimal places.
3. Underneath, the square root calculator shows the exact simplified form, whether the number is a perfect square, the two perfect squares it lies between, and the negative root.
4. The steps show the prime factorization and how the largest square factor is pulled out of the radical, so you can see why √180 becomes 6√5.

How to do square roots on a calculator

On most phone and desktop calculators, type the number first and then press the √ key: 180, then √, shows 13.4164078650. On a scientific or graphing calculator the order is usually reversed: press √ (often a second function above x²), type the number, close the bracket and press Enter. If your calculator has no √ key, raise the number to the power 0.5 with the xʸ or ^ key: 180 ^ 0.5 gives the same result. A basic calculator shows only the positive decimal root; it cannot give the exact form 6√5, which is what the steps above add.

How to simplify a radical by hand

The rule that makes simplification work is √(a × b) = √a × √b. To simplify:

1. Factor the number and look for its largest perfect-square factor. Prime factorization makes this easy, because each pair of identical primes forms a square.
2. Split the root into the square part and the rest.
3. Take the square root of the square part and write it in front.

For √180: 180 = 2² × 3² × 5. The pairs give 2 × 3 = 6, and the unpaired 5 stays inside. So √180 = √(36 × 5) = 6√5 ≈ 13.4164. The GCF calculator prints prime factorizations too, if you want to practice finding the pairs.

Estimating without a calculator

Find the two perfect squares around your number. 30 lies between 25 and 36, so √30 is between 5 and 6, a bit closer to 5. Then refine with the averaging method (also called the Babylonian or Newton’s method). Divide 30 by your guess and average the guess with the result: guess 5.5 → (5.5 + 30 ÷ 5.5) ÷ 2 = 5.47727. Repeat once more to get 5.477226. Each round roughly doubles the number of correct digits.

Square roots and simplified radicals

Three values come up again and again: √2 ≈ 1.4142, √3 ≈ 1.7321 and √5 ≈ 2.2361. Knowing them lets you estimate most simplified radicals in your head, since 5√2 ≈ 5 × 1.4142.

Where square roots are used

• Side of a square from its area. A square room of 196 sq ft is √196 = 14 ft on each side.
• Distance and diagonals. The Pythagorean theorem calculator takes a square root to find the hypotenuse, such as √(5² + 12²) = √169 = 13.
• Solving equations. The quadratic formula calculator depends on √(b² − 4ac), and x² = k gives x = ±√k.
• Statistics. A standard deviation is the square root of the variance, which brings the spread back into the original units.
• Physics and finance. Pendulum periods, free-fall times and the scaling of volatility over time all involve square roots.

Common mistakes

• Splitting a sum. √(9 + 16) is √25 = 5, not √9 + √16 = 7. The product rule works for multiplication only.
• Pulling out the wrong factor. √48 = √(16 × 3) = 4√3. Using 4 × 12 gives 2√12, which still contains a square factor.
• Forgetting the negative root when solving. x² = 64 has two solutions, x = 8 and x = −8, even though √64 is just 8.
• Treating √−9 as −3. (−3)² = 9, not −9. The square root of −9 is the imaginary number 3i.

Glossary

• Radicand: the number under the radical sign.
• Perfect square: a whole number whose square root is a whole number.
• Simplest radical form: a root with no perfect-square factor left under the radical and no radical in a denominator.
• Principal square root: the non-negative root, the value √ returns. Powers and other roots, such as cube roots, are covered by the exponent calculator.