Square Root Calculator

This square root calculator gives √x as a decimal and in simplest radical form, so √72 shows as 8.48528137 and as 6√2. It tells you whether your number is a perfect square, names the perfect squares on either side, and checks the answer by squaring it. Switch the root index for cube, 4th and 5th roots. Below: simplifying radicals, finding roots by hand, and the usual mistakes.

How to use the square root calculator

1. Type your number in the x box. The default is 72.
2. Leave the root index on square root, or choose cube root, 4th root or 5th root from the menu.
3. Read the result. For 72 the calculator shows √72 = 8.48528137 and the simplified radical 6√2, found through the largest perfect-square factor of 72, which is 36.
4. Check the context lines. 72 is not a perfect square; it sits between the perfect squares 64 and 81, so its root sits between 8 and 9. The check line squares the answer: 8.48528137² ≈ 72.
5. Open the steps to see the factoring and simplifying written out.

If you enter a negative number with the square root (or any even root) selected, the calculator explains that there is no real answer instead of showing one. The section on negative numbers below covers why.

What a square root is

The square root of a number is the value that, multiplied by itself, gives that number. √49 = 7 because 7 × 7 = 49. Every positive number has two square roots, +7 and −7 in this case, because (−7) × (−7) = 49 as well. The √ sign picks the positive one. When you need both, as when solving x² = 49, write x = ±7.

Square roots of perfect squares such as 1, 4, 9, 16 and 25 are whole numbers. Every other whole number has an irrational square root, one of the numbers that the guide to rational numbers shows cannot be written as a fraction. Its decimal never ends and never repeats. The square root of 2 is the classic case, and the same proof works for √3, √5, √72 and the rest. That is why the calculator also gives the radical form. 6√2 is exact; 8.48528137 is a rounding of it.

A square root is also a power. √x = x^(1/2), and the cube root is x^(1/3). That link is why the exponent rules apply to roots: √(x²) = (x²)^(1/2) = x for any non-negative x.

How to simplify a square root

To write a square root in simplest radical form, pull the largest perfect square you can out from under the root sign. The rule that makes this work is √(a × b) = √a × √b.

Using prime factors

When the largest square factor is not obvious, break the number into primes and take out every pair. 72 is 2 × 2 × 2 × 3 × 3. The pair of 2s comes out as one 2, the pair of 3s as one 3, and the single 2 stays inside: 2 × 3 × √2 = 6√2. The prime factorization calculator produces that list for any number. To list every factor instead, including the square ones, use the factor calculator.

A number whose only square factor is 1, such as 30, is already in simplest form: √30 cannot be simplified, and its decimal is about 5.47722558.

How to find a square root without a calculator

Two hand methods are worth knowing: a quick estimate to check an answer is in the right place, and an averaging method that closes in on any square root in a few rounds.

Estimate from the perfect squares either side

72 sits between 64 (8²) and 81 (9²), so √72 is between 8 and 9. It is 8 of the 17 steps from 64 to 81, which suggests about 8 + 8/17 ≈ 8.47. The true value is 8.485, so the estimate is close, and always slightly low, because the gaps between squares widen as numbers grow.

The divide-and-average method

Guess, divide the number by your guess, and average the guess with the result. Repeat. This is often called the Babylonian method, and each round roughly doubles the number of correct digits.

Two rounds from a rough guess give eight correct decimal places. The method works because if the guess is too big, 72 ÷ guess is too small by about the same amount, so their average lands much closer than either one.

Cube roots, 4th roots and 5th roots

The root index menu changes the question from “what times itself gives x?” to “what multiplied by itself three (or four, or five) times gives x?”

The cube root of 72 is about 4.16016765, since 4.16016765³ ≈ 72. Cube roots come up when you know a volume and need a length; the scale factor calculator uses the same idea to turn a volume ratio back into a length ratio.

Square roots of negative numbers

No real number times itself gives a negative result: a positive times a positive is positive, and so is a negative times a negative, as the negative number rules spell out. So √−16 has no real answer. Mathematicians write it as 4i, using the imaginary unit i, but that is outside what this calculator computes, and it shows a message instead. Odd roots are different: (−2)³ = −8, so the cube root of −8 is the real number −2.

Square roots of decimals and fractions

The same rules apply. √0.25 = 0.5 because 0.5 × 0.5 = 0.25, and √1.44 = 1.2. Notice that the square root of a number between 0 and 1 is bigger than the number itself: √0.9 ≈ 0.9487. For a fraction, take the root of the top and the bottom separately: √(9/16) = 3/4.

Where square roots come up

Square roots appear whenever you know an area or a squared quantity and need the length behind it.

• Side of a square from its area. A square room of 144 square feet is √144 = 12 feet on each side. A square patio of 200 square feet is √200 ≈ 14.14 feet on each side.
• The diagonal of a rectangle. By the Pythagorean theorem, a 3 by 4 rectangle has a diagonal of √(3² + 4²) = √25 = 5. A 9 by 12 foot room has a diagonal of √(81 + 144) = √225 = 15 feet.
• Scaling areas back to lengths. If one square plot has 2.25 times the area of another, its sides are √2.25 = 1.5 times as long.

In each case, do the arithmetic inside the root first, following the order of operations, and only then take the root.

Common mistakes

• Splitting a root across addition. √(9 + 16) is √25 = 5, not 3 + 4 = 7. The product rule works for multiplication only.
• Stopping before the radical is fully simplified. 2√18 is correct but not simplest; it is 6√2.
• Dropping the ± when solving an equation. x² = 49 has two solutions, 7 and −7. The √ symbol on its own means only the positive one.
• Confusing √x with x ÷ 2. √16 is 4, which happens to equal 16 ÷ 4, but √36 is 6, not 18.
• Assuming √(x²) = x for negative x. √((−5)²) = √25 = 5. The correct rule is √(x²) = |x|, the absolute value of x.
• Rounding too early. Keep 6√2 exact through a calculation and round only the final decimal, with the rounding calculator if you need a set number of places.

Frequently asked questions

Related calculators

Guides that use this calculator

Longer explanations of the method behind it, with worked examples.