Root Calculator

What an nth root means and how the root calculator finds it

Raising a number to a power multiplies it by itself a set number of times: 4 squared (4 raised to the power of 2) is 4 x 4 = 16. A root undoes that operation. Instead of starting with the base and finding the power, a root starts with the result and asks what base produced it. The square root of 16 is 4, because 4 is the number that, squared, gives back 16. Written as an equation, taking the nth root of a number is the same operation as raising it to the power of 1/n:

The same idea extends to any degree. The cube root of a number asks what base, raised to the power of 3, gives that number back. Take 27: since 3 x 3 x 3 = 27, the cube root of 27 is 3. A fourth root works the same way one step further: it asks what base, raised to the power of 4, gives that number back, and so on for any whole-number degree. In every case, the root degree tells you how many times the answer needs to be multiplied by itself to land back on your original number.

Most numbers do not have a tidy whole-number root. The square root of 2, for example, is an irrational number, its decimal expansion runs on forever without repeating, starting 1.41421356. This calculator rounds a result like that to 10 significant figures so it displays cleanly, rather than trailing off into the binary floating-point noise a computer would otherwise produce.

A few more square roots worked out for quick reference: the square root of 3 (often typed as sqrt 3) is approximately 1.7320508, another irrational value like the square root of 2 above. The square root of 13 is approximately 3.6055513, and the square root of 24 is approximately 4.898979, both also irrational. Not every square root runs on forever, though: the square root of 400 is exactly 20, a clean perfect-square example, since 20 multiplied by 20 comes to exactly 400 with nothing left over.

Square root table for the numbers people look up most

The table lists each square root to four decimal places, next to its simplified radical form. Simplifying means pulling the largest perfect square out from under the radical sign: 24 is 4 x 6 and the square root of 4 is 2, so the square root of 24 is 2√6. A number with no square factor, like 29, is already as simple as it gets. The decimals come from the same code as the calculator above, so any row can be checked against it.

A few names are worth untangling. Root 2 and root 3 are the spoken forms of the square root of 2 and the square root of 3, and root 2 is about 1.4142, the diagonal of a square with sides of 1. Under root is a regional way of saying the same thing, so under root of 3 is simply the square root of 3, about 1.7321. And what is the square root of 4? It is 2, because 2 x 2 is 4. Strictly, -2 also squares to 4, but the radical sign always means the positive answer, called the principal root.

What is the square root of a square? Squaring and rooting undo each other, with one catch: squaring throws away the sign, so the root of a square is always the positive value, the absolute value of the original number.

The square root of (-5) squared is therefore 5, not -5. The same rule explains why the calculator shows only 5 for the square root of 25, even though -5 also squares to 25.

Higher roots follow the same pattern. Every root of 1 is 1, so the cube root of 1 is 1, while the cube root of 250 is about 6.2996, because 6.2996 x 6.2996 x 6.2996 comes back to 250. A negative number can have a real cube root but never a real square root.

Why negative numbers behave differently depending on the degree

Whether a negative number has a real root depends entirely on whether the root degree is even or odd, and the reason comes straight from how multiplying negative numbers works. A negative number multiplied by itself an even number of times always ends up positive, because the negative signs cancel out in pairs. That means there is no real number which, squared (an even degree of 2), produces a negative result, so an even-degree root of a negative number simply does not exist among the real numbers. This tool reports that case as an error rather than guessing at an answer.

An odd-degree root of a negative number is a different story. Multiplying a negative number by itself an odd number of times leaves one negative sign unpaired, so the result stays negative. Take the cube root of -8: -2 x -2 x -2 works out to -8, so the cube root of -8 is a perfectly real -2. Note that JavaScript's built-in power function cannot compute this directly for a fractional exponent of a negative base, so this calculator handles the odd, negative case explicitly by finding the root of the positive magnitude first and then reapplying the negative sign, which is the mathematically correct approach.

This even-versus-odd rule is worth remembering on its own: a square root, a 4th root, a 6th root, or any other even degree, of a negative number, has no real answer. A cube root, a 5th root, a 7th root, or any other odd degree, of a negative number, always has a real answer, and that answer is always negative.

The square root sign itself, √, is called a radical sign, and the number underneath it is the radicand. A radical calculator is really just another name for this same square root or nth root operation, and a small raised number in front of the radical sign (like a tiny 3 for a cube root) marks the root degree the same way the degree field above does.

Estimating a square root by hand

Long before calculators, mathematicians narrowed in on a square root using a simple repeated-averaging method, sometimes called the Babylonian method after its earliest known use. It only needs a starting guess and basic division, and it gets close to the true answer within three or four rounds:

1. Make a reasonable starting guess, b, for the square root of your number, a.
2. Divide a by b to get c. If c is already as precise as you need, stop, since a good guess divided into a makes c and b converge on the same value.
3. Average b and c, and use that average as your new guess.
4. Repeat the division step with the new guess until the answer stops changing.

In symbols, one round of that averaging is:

Worked example: the square root of 27, to 3 decimal places. Since 5² is 25 and 6² is 36, a good starting guess is 5.125 (close to 5, but nudged toward 27 being nearer 25 than 36).

• 27 ÷ 5.125 = 5.268. Average of 5.125 and 5.268 = 5.197.
• 27 ÷ 5.197 = 5.195. Average of 5.197 and 5.195 = 5.196.
• 27 ÷ 5.196 = 5.196. The guess and the division agree: converged.

The square root of 27 is approximately 5.196, matching what the calculator above gives instantly. Choosing a closer starting guess reaches that answer in fewer rounds, but even a rough first guess converges quickly, since each round roughly doubles the number of correct digits.

Estimating an nth root by hand

The same averaging idea extends to any root degree, not just square roots, using a weighted average instead of a plain one. To estimate the nth root of a number a:

1. Make a starting guess, b.
2. Divide a by b raised to the power of (n − 1), to get c. If c is precise enough, stop.
3. Take a weighted average: multiply b by (n − 1), add c, then divide the total by n. Use that as your new guess.
4. Repeat the division step until the answer stops changing.

As one line, the weighted average for a root of degree n is:

Worked example: the 8th root of 15, to 3 decimal places. Starting from a guess of 1.432, and using b^7 (b raised to the power of 7, since the root degree here is 8 and n − 1 = 7):

• 15 ÷ 1.432^7 = 1.215. Weighted average: (1.432 × 7 + 1.215) ÷ 8 = 1.405.
• 15 ÷ 1.405^7 = 1.388. Weighted average: (1.405 × 7 + 1.388) ÷ 8 = 1.403.
• 15 ÷ 1.403^7 = 1.402. Weighted average: (1.403 × 7 + 1.402) ÷ 8 = 1.403. Converged.

The 8th root of 15 is approximately 1.403, again matching the calculator above. This weighted-average method is the general case behind the plain averaging method used for square roots, since a square root is simply the 2nd root, where the weighting reduces to a plain average of two numbers. A full worked walkthrough of the square-root case, plus the alternative long-division method for finding a root digit by digit, is in how to calculate a square root by hand.