Root Calculator
Calculate square roots, cube roots, and nth roots of any number.
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About
Root Calculator
The nth root of a number a is the number b that gives you a when it is used n times as a factor. In symbols:
ⁿ√a = b, which is the same statement as bⁿ = a
So the fourth root of 64 is 2.8284271247462, because that number multiplied by itself four times comes back to 64. Every root question is a power question read backwards, and that is the single most useful thing to know about them. If you can square a number you can already check a square root; if you can cube one you can check a cube root.
The parts of a radical
The tick-and-bar symbol is the radical. The number sitting under the bar is the radicand. The small number tucked into the notch is the index, and it says how many times the answer is used as a factor. A square root has an index of 2, which is the one case where nobody writes it: the plain radical always means index 2.
The two most common roots have names. Index 2 is the square root, index 3 is the cube root. Above that they are read off the ordinal, so index 4 is the fourth root and index 8 is the eighth root. There is no special word for them, which is exactly why the general root calculator on this page has an index field.
The symbol has a traceable history. Christoff Rudolff put the radical sign into print in his 1525 algebra book Die Coss, using a mark resembling a lowercase r for radix, the Latin for root. It had no bar over the top, so the reader had to guess where the radicand ended. Descartes added the horizontal bar, the vinculum, in La Geometrie in 1637, and the notation has barely changed since.
Roots written as exponents
A root is a fractional power. This is not an analogy, it is the definition that mathematics actually uses:
ⁿ√a = a^(1/n)
Which is why every scientific calculator can produce a seventh root without having a seventh-root button. It computes 15^(1/7). Once roots are written this way the ordinary rules of exponents take over. A cube root of a square is a^(2/3). Multiplying roots of the same index just multiplies the radicands, because a^(1/n) x b^(1/n) = (ab)^(1/n). The rule that trips people is addition, which has no such shortcut. The square root of 9 plus 16 is 5, while the square root of 9 plus the square root of 16 is 7. Roots do not distribute over addition, ever.
The principal root, and the answer people expect
Ask for the square root of 16 and this calculator returns 4. Ask which numbers square to 16 and the answer is 4 and -4. Both are true and they are not the same question. The radical symbol is defined to return the principal root, the non-negative one, so that the expression names exactly one number. Solving an equation is where the second root reappears: x² = 16 has the two solutions x = 4 and x = -4, and that is why the plus-or-minus sign is written by hand in the quadratic formula rather than being carried inside the radical.
Odd indexes behave differently and more simply. The cube root of -27 is -3, a perfectly ordinary real number, because multiplying three negatives leaves a negative. Any odd root of a negative number has a real answer. Any even root of a negative number does not, since nothing real can square to something negative. The convention there is to define i as the number whose square is -1 and write the answer as 4i for the square root of -16. This calculator says so rather than returning an error, because the value is well defined once you accept complex numbers.
Perfect powers
Some radicands come out whole. The perfect squares below 200 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 and 196. The perfect cubes below 1,000 are 1, 8, 27, 64, 125, 216, 343, 512 and 729. Both lists contain 64, which is 8 squared and 4 cubed, and is a sixth power for the same reason. Recognising these on sight makes estimation much faster, because you can place any radicand between two you already know: 50 sits between 49 and 64, so its square root is a shade over 7.
Everything else gives an irrational answer, meaning the decimal never terminates and never repeats. The square root of 2 is the famous case and the proof that it cannot be written as a fraction is one of the oldest results in mathematics, credited to the Pythagoreans. The story that Hippasus was drowned at sea for revealing it is a much later legend rather than recorded history.
Simplifying a square root
An irrational root can often be tidied without losing any accuracy. Pull out the largest square factor:
√72 = √(36 x 2) = 6√2
The result panel shows this simplified form whenever the radicand is an integer with a square factor. It matters in algebra and trigonometry, where an exact answer is wanted rather than a decimal, and it is the reason the sine of 45 degrees is written as the square root of 2 over 2 instead of 0.7071. The related habit is rationalising the denominator, replacing 1 over the square root of 2 with the square root of 2 over 2, which is the same number written so the messy part sits on top.
Estimating a square root by hand
The method below needs nothing beyond division and averaging. To find the square root of a:
- Estimate a number b.
- Divide a by b. Call the result c. If it agrees with b to as many decimal places as you need, stop.
- Average b and c, and use that as the new guess.
- Go back to step two.
Finding the square root of 27 to three decimals, starting from a guess of 5.125: 27 divided by 5.125 is 5.268, and the average of 5.125 and 5.268 is 5.197. Then 27 divided by 5.197 is 5.195, and the average of those is 5.196. One more pass returns 5.196 again, so the answer is 5.196.
The logic is simple. If your guess is too small, dividing gives something too large, and the truth is caught between them. Averaging steps into the middle, and each pass roughly doubles the number of correct digits. This is Heron's method, described by Heron of Alexandria in the first century, though a Babylonian clay tablet catalogued as YBC 7289 already records the square root of 2 as 1.41421296 in base 60, correct to about six decimal places, more than three thousand years ago.
Estimating any nth root
The same idea works for a higher index with two changes. Divide by b raised to n-1 rather than by b, and take a weighted average that leans on the old guess:
new estimate = [b x (n - 1) + c] ÷ n, where c = a ÷ bⁿ⁻¹
Finding the eighth root of 15, starting from a guess of 1.5: 1.5 to the seventh power is 17.0859, and 15 divided by that is 0.8779. The weighted average is (1.5 x 7 + 0.8779) / 8, which is 1.4222. Next pass: 1.4222 to the seventh is 11.7709, 15 divided by that is 1.2743, and the weighted average is 1.4038. A third pass gives 1.4029, and the true value is 1.402851, so the eighth root of 15 is 1.403 to three decimals.
The weighting is what keeps it stable. A plain average of b and c overshoots badly once the index is large, because raising to the seventh power exaggerates every error. Giving the old guess seven of the eight votes damps that down. This procedure is Newton's method applied to xⁿ - a, written out without any calculus, and the step table on this page runs exactly these lines so you can follow the convergence.
Where roots turn up
Square roots sit inside the distance formula and the Pythagorean theorem, so any diagonal measurement produces one. Standard deviation is a square root, which is the only reason it can be reported in the same units as the data instead of in units squared. Higher roots appear whenever a total growth figure has to be spread evenly across periods: the compound annual growth rate of an investment is the nth root of the ending value over the beginning value, minus one, where n is the number of years. An account that goes from 10,000 to 18,000 over 5 years has grown at the fifth root of 1.8, minus 1, which is 12.47% a year.
How to use this page
The square root and cube root calculators take one number. The general root calculator takes an index as well, so set it to 2, 3 or anything else, including a decimal. Each result panel shows the answer to 14 significant figures, the check line raising it back to the power, and the exponent form. Below each answer, the step by step solution runs the hand method from a sensible starting guess and prints every division and average, ending with the verification. When the radicand is a perfect power the first estimate is already exact and one pass confirms it.
Common questions
Frequently asked questions
A square root asks which number used twice as a factor gives the radicand, so the square root of 16 is 4 because 4 x 4 = 16. A cube root asks for the number used three times, so the cube root of 27 is 3 because 3 x 3 x 3 = 27. The index is the only thing that changes, and the general root calculator handles any value of it.
The radical symbol is defined to return the principal root, which is the non-negative one, so the expression names a single number. The two-answer version belongs to equations: x squared = 16 has the solutions 4 and -4. That is why the quadratic formula writes the plus-or-minus sign in front of the radical instead of leaving it implied.
Not within the real numbers, because no real number squares to a negative. Mathematics defines i as the number whose square is -1, which makes the square root of -16 equal to 4i. This calculator returns that form rather than an error. Odd indexes are different: the cube root of -27 is -3, an ordinary real number, since three negative factors leave a negative product.
The nth root of a is a raised to the power 1/n. A square root is the 1/2 power, a cube root is the 1/3 power. Combining the two, the cube root of a squared is a to the 2/3. This is how calculators and spreadsheets compute roots at all, and in Excel or Google Sheets you would type =64^(1/4) to get the fourth root of 64.
Fourteen significant figures, with trailing zeros trimmed, which is close to the limit of standard double-precision arithmetic. Irrational roots such as the square root of 2 have decimal expansions that never end, so any printed value is a truncation. The check line squares or cubes the answer back so you can see how closely it returns the radicand.
It means pulling the largest square factor out from under the radical, so the square root of 72 becomes 6 times the square root of 2. The value is identical, but the simplified form is exact rather than a decimal and is what algebra and trigonometry courses ask for. The result panel shows it whenever the radicand is an integer that has a square factor.
Guess, divide, average, repeat. For a square root, divide the radicand by your guess and average the two numbers; the true value always lies between them, so each pass narrows the gap and roughly doubles the correct digits. For a higher index, divide by the guess raised to n-1 and take the weighted average [b(n-1)+c]/n. The step table on this page prints exactly these lines.
A number whose root is a whole number. The perfect squares start 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100; the perfect cubes start 1, 8, 27, 64, 125 and 216. Knowing them lets you bracket any other radicand: 50 falls between 49 and 64, so its square root is a little above 7. The result panel flags a radicand that is an exact power.