Square Root Calculator

What is a square root?

The square root of a number n is the value that, when multiplied by itself, equals n. It is written √n, and the bar symbol (√) is called a radical. For example, the square root of 25 is 5, because 5 × 5 = 25.

A number is a perfect square when its square root is a whole number — 1, 4, 9, 16, 25 and 36 are all perfect squares. Most numbers are not: √2, √3 and √10 are irrational, meaning their decimals never terminate or repeat. This square root calculator handles both cases, returning the exact decimal for any number and a tidy simplified radical for whole numbers.

How does the calculator work?

The tool does two things at once. First it computes the decimal value using standard square-root math and rounds it to six decimal places. Then, for whole-number inputs, it simplifies the radical by pulling out the largest perfect-square factor.

The method for the radical form is:

√n = √(largest perfect square × remainder) = (√perfect square) × √remainder

To find that largest perfect square, the calculator checks each integer i from √n downward and stops at the first one whose square divides n evenly. The result is reported as a√b, where a is the part taken outside the radical and b is whatever stays inside.

How do you simplify a square root step by step?

Simplifying a radical means rewriting it so the number under the root has no perfect-square factors left. Follow these four steps:

1. Find the largest perfect square that divides your number (4, 9, 16, 25, 36, …).
2. Split the number into that perfect square multiplied by the remainder.
3. Take the square root of the perfect square and move it outside the radical.
4. Leave the remainder under the radical sign.

Worked example 1: √72

• 72 = 36 × 2, and 36 is the largest perfect-square factor.
• √72 = √36 × √2 = 6√2
• As a decimal: √72 ≈ 8.485281

Worked example 2: √200

• 200 = 100 × 2, and 100 is the largest perfect-square factor.
• √200 = √100 × √2 = 10√2
• As a decimal: √200 ≈ 14.142136

Worked example 3: √48

• 48 = 16 × 3, and 16 is the largest perfect-square factor.
• √48 = √16 × √3 = 4√3
• As a decimal: √48 ≈ 6.928203

You can reproduce any of these by typing the number into the calculator above and pressing Calculate √.

Perfect squares reference table (1–15)

Memorising the first fifteen perfect squares makes simplifying radicals far faster, because you can spot the largest square factor at a glance.

Number (n)	Square (n²)	√(n²)
1	1	1
2	4	2
3	9	3
4	16	4
5	25	5
6	36	6
7	49	7
8	64	8
9	81	9
10	100	10
11	121	11
12	144	12
13	169	13
14	196	14
15	225	15

When the number under a radical appears in the middle column above, the root is a whole number and the radical is a perfect square.

What are square roots used for in real life?

Square roots show up far beyond the maths classroom — anywhere a quantity grows with the area or square of another.

• Geometry and area. If a square room has an area of 144 square feet, each side is √144 = 12 feet. Square roots reverse any area calculation back into a length.
• The Pythagorean theorem. For a right triangle, the hypotenuse is c = √(a² + b²). A triangle with legs of 3 and 4 has a hypotenuse of √(9 + 16) = √25 = 5 — the classic 3-4-5 triangle. Legs of 6 and 8 give √(36 + 64) = √100 = 10.
• Standard deviation. In statistics, the standard deviation is the square root of the variance, which is why spread is reported in the same units as the data.
• Distance and physics. The straight-line distance between two points, the speed of a falling object, and the period of a pendulum all rely on square roots.

For everyday percentage work alongside these calculations, the percentage calculator is a handy companion, and the BMI calculator is one common formula where a value is divided by a squared height.

Tips and common mistakes

• √(a + b) is not √a + √b. This is the single most frequent error. √(9 + 16) = √25 = 5, not 3 + 4 = 7. The root applies to the whole sum.
• Always pull out the largest perfect square. Writing √72 as 2√18 is correct arithmetically but not fully simplified, because 18 still contains the factor 9. The simplest form is 6√2.
• Check your simplification by squaring back. (6√2)² = 36 × 2 = 72, confirming the answer.
• Use the table. Recognising 36 as a factor of 72 instantly is the fastest route to 6√2.

What about negative numbers and accuracy?

A negative number has no real square root, because any real number multiplied by itself is zero or positive — there is no real value whose square is, say, -9. Roots of negatives are imaginary numbers (written with the symbol i), which fall outside this tool’s scope. The calculator only accepts inputs of 0 or greater and will show an error otherwise.

For accuracy, remember that irrational roots like √2 and √3 never end. The decimal shown is rounded to six places, so treat it as a precise approximation rather than an exact value; whenever an exact answer matters, keep the radical form (such as 6√2) instead of the decimal.

This calculator is an educational and reference tool. For graded coursework or engineering work, double-check results and follow the rounding rules your instructor or specification requires.