Square Root Calculator

About the Square Root Calculator

The square root of a number is the value that, multiplied by itself, gives that number. The root of 144 is 12, because 12 × 12 = 144.

Most numbers are not that obliging. The root of 72 is 8.485281374…, continuing forever without repeating. But 72 does have a tidier form: it is exactly 6√2. That is not an approximation — it is the true value, written in a way that fits on a line.

This calculator gives both. The decimal, rounded to whatever precision you ask for, and the exact simplified surd where one exists. For anything beyond rough arithmetic, the exact form is the one to carry forward, because rounding early is how error creeps into a long calculation.

How to Use the Square Root Calculator

Find the square root takes a number and returns its principal root. Set the decimal places if you need more or fewer than six.

Square a number goes the other way, multiplying a number by itself. Useful for checking a root, and for the geometry cases where squaring is the operation you actually want.

Press Calculate. You get the answer, the exact form, and a note on whether the number is a perfect square or the root is irrational.

Negative inputs are refused in root mode, and the reason is explained rather than returned as an error code: no real number squares to a negative.

How Square Roots Are Calculated

Conceptually, finding a square root means asking what number squares to the one you have. Mechanically there are two useful things going on.

Perfect squares are found by recognition. 144 is 12², so the root is 12 exactly. The calculator checks this first, because when it applies there is nothing else to do.

Simplifying a surd means pulling out the largest perfect-square factor. For 72:

72 = 36 x 2
√72 = √36 x √2 = 6√2

This works because √(ab) = √a × √b. Any factor that is itself a perfect square comes out from under the radical as a whole number. The method is to test divisibility by 4, 9, 16, 25 and so on, taking the largest that divides.

Getting the largest factor matters. For 288 you could notice 4 × 72 and write 2√72 — correct, but unfinished, because 72 still contains 36. Taking 144 × 2 gives 12√2 in one step.

Estimating a decimal starts by bracketing. The root of 50 lies between 7 and 8, because 7² = 49 and 8² = 64, and much closer to 7. From there the Babylonian method converges fast: guess, divide the number by your guess, average the two, repeat. Starting at 7, you get 50/7 ≈ 7.143, average 7.071 — already correct to four decimal places after one step.

Square Root Formula

√(a x b) = √a x √b
√(a / b) = √a / √b
√(a²b)   = a√b            the simplification rule
√x x √x  = x
(√x)²    = x              for x >= 0
√(x²)    = |x|            note the absolute value

That last line catches people out. √(x²) is not x — it is the absolute value of x, because the root symbol always returns the non-negative option. √((−5)²) is √25, which is 5, not −5.

The distinction between the two roots matters when solving equations:

√9 = 3                    the principal root, one value
x² = 9  ->  x = ±3        an equation, two solutions

The symbol means one thing; the equation asks a different question.

Note also what does not work:

√(a + b) ≠ √a + √b

√(9 + 16) is √25 = 5, while √9 + √16 is 3 + 4 = 7. The radical does not distribute over addition, only over multiplication and division.

Step-by-Step Example

Simplify √72.

Step 1 — look for perfect-square factors. 72 divides by 4, by 9, and by 36. The largest is 36.

Step 2 — split it. 72 = 36 × 2.

Step 3 — take the root of each part. √36 = 6, and 2 has no square factor, so it stays under the radical.

Step 4 — write it out. √72 = 6√2.

Step 5 — check numerically. √2 ≈ 1.414214, and 6 × 1.414214 ≈ 8.485284. The calculator's decimal answer is 8.485281, which agrees to the precision of the rounding.

Now estimate √50 without a calculator.

Step 1: bracket it.  7² = 49, 8² = 64  ->  just above 7
Step 2: divide.      50 / 7 = 7.142857
Step 3: average.     (7 + 7.142857) / 2 = 7.071429

The true value is 7.071068. One round of the method got within 0.0004, which is plenty for a mental estimate and rather satisfying for arithmetic that is four thousand years old.

Understanding Your Result

The answer is the principal root, rounded to your chosen precision.

The exact form is the value written without rounding. For a perfect square it is a whole number. For anything else it is a surd such as 6√2, which is exact where the decimal is not. If you are carrying the value into further work — particularly geometry or algebra — use this form and convert to a decimal only at the end.

The type line says whether the number is a perfect square or the root is irrational. Irrational means the decimal never terminates and never repeats, which is not a limitation of the calculator but a property of the number. √2 has been known to be irrational for around 2,500 years.

One thing the result does not show: the negative root. √144 is 12, and −12 also squares to 144. The symbol conventionally means the non-negative one. If you are solving x² = 144, remember there are two answers.

When Should You Use This Calculator?

Geometry. The side of a square from its area, the diagonal of a rectangle, the hypotenuse of a right triangle — all square roots.

Pythagoras. c = √(a² + b²) is the most common square root in practical use, from carpentry to navigation.

Statistics. Standard deviation is the square root of the variance, which is why it shares the units of the original data and variance does not.

Physics. Free-fall times, orbital periods, root-mean-square values and anything involving an inverse-square law.

Algebra homework. Questions that ask for surd form specifically, where a decimal answer scores nothing.

Quadratic equations. The formula contains √(b² − 4ac), and whether that is positive, zero or negative determines how many solutions exist.

Common Mistakes

Assuming a negative number has a real root. √(−9) does not exist among the real numbers. It is 3i in complex arithmetic, which is a different subject.

Distributing the root over addition. √(a + b) is not √a + √b. This is probably the most common algebra error involving radicals.

Stopping the simplification too early. 2√72 is correct but unfinished. Always take the largest perfect-square factor.

Forgetting the second root when solving an equation. x² = 25 has two solutions, +5 and −5. Reporting only 5 loses half the answer.

Writing √(x²) as x. It is |x|. For positive x they agree; for negative x they do not.

Rounding the surd too early. Working with 8.485 instead of 6√2 through several steps accumulates error. Keep the exact form until the last moment.

Confusing the square root with halving. The root of 16 is 4, not 8. The two coincide only at 4, which is occasionally just enough to hide the mistake.