The square root calculator above gives you two answers for the price of one: the decimal value, and the exact simplified radical form. Most online calculators stop at the decimal, which is fine for measuring something and useless for algebra, where √72 needs to stay as 6√2 rather than becoming 8.485281.
Arb Digital publishes this as part of a free maths toolkit. The simplification step is the part that gets marked in exams and the part that keeps later arithmetic exact, so this tool shows the factorisation it used rather than just handing over the answer.
What This Square Root Calculator Does
Enter a number and the tool returns its principal square root to the precision you choose. Alongside it you get the simplified radical form — the exact expression with the largest possible perfect square pulled outside the radical sign — the negative root, the square of your original number, and the two perfect squares your number sits between, which is the fastest way to sanity-check a decimal result.
An optional multiplier field handles expressions such as 3√72, which turn up constantly in geometry, trigonometry and physics. The tool folds the multiplier into the simplified form as well, so 3√72 is reported as 18√2 rather than as a decimal that loses the exact structure.
Negative inputs are handled explicitly. The square root of a negative number is not a real number, and the calculator says so and names the imaginary form rather than returning a silent error.
How to Use It
- Enter the number under the radical. Whole numbers give the cleanest simplified forms; decimals are supported and return decimal answers.
- Set the decimal places. Six is a sensible default. Increase it when the root feeds into a further calculation, since rounding early compounds the error.
- Add a multiplier if you need one. Enter 3 to evaluate 3√72. Leave it at 1 for a plain root.
- Read the simplified radical. This is the exact answer. Use it in algebra and keep the decimal only for the final measurement.
- Check the nearest perfect squares. If your decimal answer does not sit between their roots, something in the input is wrong.
The Formula / How It's Calculated
The square root of n is the number that, multiplied by itself, gives n: √n = x means x × x = n. Since 6 × 6 = 36, √36 = 6. It is also the exponent form n1/2, which is why the exponent calculator produces the same answer when you set the power to 0.5.
Simplification works by factoring out the largest perfect square that divides your number. For the loaded example of 72: the perfect squares that divide 72 are 4, 9 and 36, and the largest is 36. So √72 = √(36 × 2) = √36 × √2 = 6√2. The decimal 8.485281… is the same value, but 6√2 is exact and stays exact through every later step. The identity that makes this legal is √(a × b) = √a × √b for non-negative a and b, described in MathWorld's entry on the square root.
Numerically, the value itself is computed by the browser's built-in square-root routine, which uses a hardware instruction correctly rounded to the nearest representable double under the IEEE 754 standard. That is more accurate than any hand method, though the digits still stop at about 16 significant figures.
Why the Square Root of a Negative Number Is Not Real
Multiplying two positive numbers gives a positive result. Multiplying two negative numbers also gives a positive result. There is no real number that produces a negative when multiplied by itself, so √(−9) has no real value. This is not a limitation of the calculator — it is a property of the real number line.
Mathematics extends the number system to handle it. The imaginary unit i is defined so that i2 = −1, which makes √(−9) = 3i. That is a complete and consistent answer, used constantly in electrical engineering and signal processing, but it lives in the complex numbers rather than the reals. The calculator reports the imaginary form so you know the answer exists, while making clear it is not a real number.
This is exactly where square roots and cube roots part company. The cube root of a negative number is real, because a negative multiplied by itself three times stays negative: (−2)3 = −8, so the cube root of −8 is −2. Odd roots of negatives are real; even roots are not. Our cube root calculator handles that case properly.
Every Positive Number Has Two Square Roots
Both 5 and −5 square to 25, so both are square roots of 25. The radical symbol √ is defined to mean the principal — that is, non-negative — root only, so √25 = 5 and not ±5. When you solve an equation rather than evaluate a symbol, both roots matter: x2 = 25 has the solutions x = 5 and x = −5.
Dropping the negative solution is one of the most common algebra errors, and it changes answers in quadratics, in distance problems and in physics questions where a negative value has a physical meaning such as direction. The rule of thumb: √ gives one value, but solving a squared equation gives two. The quadratic equation solver makes the ± explicit, since the quadratic formula carries a square root at its heart.
Estimating a Square Root Without a Calculator
Knowing the perfect squares up to 400 lets you bracket any root in seconds. To estimate √72: 82 = 64 and 92 = 81, so the answer is between 8 and 9. 72 is closer to 64 than to 81, so the answer is a little under halfway — around 8.5. The true value is 8.4853, so a five-second estimate was accurate to about half a percent.
You can refine it with one round of the ancient divide-and-average method, still the fastest hand technique there is. Guess 8.5, divide 72 by 8.5 to get 8.4706, then average the guess and the quotient: (8.5 + 8.4706) ÷ 2 = 8.4853. One iteration from a rough guess gives four correct decimal places. This method roughly doubles the number of correct digits each time you repeat it, which is why it survived for millennia before electronic calculators.
Rational, Irrational, and Why It Matters
The square root of a perfect square is a whole number. The square root of any other whole number is irrational — it never terminates and never repeats, so no decimal you write down is ever exactly right. √2 = 1.41421356… continues forever, which is why keeping the radical form matters when precision is required.
This has a practical consequence in geometry. The diagonal of a 1 × 1 square is exactly √2, and the diagonal of any square with side s is s√2 — a direct consequence of the relationship described in MathWorld's entry on the Pythagorean theorem. Writing the answer as 1.414s introduces an error immediately; writing it as s√2 keeps it exact until the final step. The same appears in the 45-45-90 triangle and throughout the Pythagorean theorem calculator, where hypotenuse lengths are square roots far more often than they are whole numbers. Rounding only at the very end, never in the middle, is the single habit that keeps geometry answers accurate.
Square Roots of Fractions and Decimals
For a fraction, take the root of the top and the bottom separately: √(9/16) = √9 ÷ √16 = 3/4. If the denominator is not a perfect square, the traditional next step is rationalising it — multiplying top and bottom by the radical so the denominator becomes whole. √(1/3) becomes √3 ÷ 3, which is easier to evaluate by hand and is still the expected form in most exam schemes. Our fraction calculator handles the surrounding fraction arithmetic.
Decimals catch people out because the root of a number below 1 is larger than the number itself. √0.25 = 0.5, and √0.01 = 0.1. Squaring shrinks values between 0 and 1, so taking a root of them grows them back. If your input is less than 1 and your answer is smaller than your input, the calculation has gone the wrong way.
Arb Digital publishes hundreds of free calculators and converters. Every one is free, needs no account, and runs entirely in your browser.
Browse All Free Tools Request a ToolCommon Mistakes to Avoid
- Forgetting the negative solution when solving x2 = k — the equation has two answers even though the radical symbol returns one.
- Splitting √(a + b) into √a + √b — the product rule does not apply to sums. √(9 + 16) is 5, not 3 + 4.
- Leaving a non-maximal square inside the radical — √72 written as 2√18 is correct but not simplified; the largest square factor is 36.
- Rounding the root before using it again — errors compound quickly, so keep the exact radical form until the final step.
- Assuming the root of a decimal is smaller than the decimal — below 1, roots grow: √0.25 is 0.5.
Related Free Tools From Arb Digital
Square roots sit at the centre of the Pythagorean theorem calculator and the quadratic equation solver, both of which return radical results. For odd roots use the cube root calculator, for the general case use the exponent calculator with a power of 0.5, and for fractional radicands the fraction calculator keeps the arithmetic exact. The free online tools hub lists the whole set.
Frequently Asked Questions
Find the largest perfect square that divides the number, take its root outside the radical, and leave the rest inside. For 72 the largest square factor is 36, so the square root of 72 simplifies to 6 times the square root of 2.
There is no real answer, because no real number multiplied by itself gives a negative result. In complex numbers the square root of −9 is 3i, using the imaginary unit i defined so that i squared equals −1.
Every positive number has two, one positive and one negative, since both square to the same value. The radical symbol refers only to the positive or principal root, so you must add the negative solution yourself when solving an equation.
No. It cannot be written as a fraction of two whole numbers, and its decimal expansion never terminates or repeats. Only the roots of perfect squares are whole numbers; every other whole number has an irrational root.
Bracket it between the two nearest perfect squares, guess a value between them, divide the original number by your guess, then average the guess and the quotient. One round of this usually gives three or four correct decimal places.
Because squaring a number between 0 and 1 makes it smaller, so the inverse operation makes it larger. The square root of 0.25 is 0.5, and the square root of 0.01 is 0.1.
Yes, take the root of the numerator and denominator separately. The square root of 9 over 16 is 3 over 4. If the denominator is not a perfect square, rationalise it by multiplying the top and bottom by the radical.
This tool is provided for general educational use. Decimal results are limited by double-precision arithmetic, so use the simplified radical form when an exact value is required.
