Davided.

Square Root Calculator

Find the square root of any number — plus nth root, cube root, decimal precision control, and simplified radical form (e.g. √72 = 6√2).

✓ Last reviewed: Sources: OpenStax Algebra 1 — 5.2.2 The nth Root

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The number whose root you want to find.

2 = square root, 3 = cube root, and so on.

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How many decimal places to show for non-exact results.

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Square root calculator: square root, cube root and nth root, simplified

The Davided square root calculator finds the square root of any number by default, but it isn’t limited to that: change the “root degree” field and it becomes a cube root calculator, a fourth-root calculator, or any nth root calculator you need, up to degree 20. Type a number, leave the degree at 2 for a square root, and the result appears instantly — as a decimal, and, when the number allows it, as a simplified radical.

That simplified radical form is the calculator’s main advantage over most square root tools online: instead of just handing you a decimal, it also shows you that √72 = 6√2, or that ∛16 = 2∛2, fully worked out. You also control the decimal precision (0 to 10 places) and get an automatic “power check” that verifies the answer by raising it back to the original degree.

This calculator is built for algebra homework, checking simplification steps by hand, engineering and geometry calculations (diagonals, areas, volumes), and any situation where a decimal alone doesn’t tell the whole story.

How to use the square root calculator

  1. Enter the number. This is the “radicand” — the number under the root sign. It can be a whole number, a decimal, zero, or negative.
  2. Set the root degree. Leave it at 2 for a standard square root, set it to 3 for a cube root, or use any whole number from 1 to 20 for other roots.
  3. Read the result. You get the decimal value, and, whenever the radicand is a whole number that simplifies, the radical form (like 6√2) alongside it.
  4. Adjust precision if needed. Open Advanced options to change how many decimal places are shown — the default is 6.

If the root degree is even and the number is negative, there’s no real-number answer, and the calculator tells you so instead of returning a value (more on why below).

How the calculator works

The principal square root

The square root of a number n is the non-negative value that, multiplied by itself, gives n. For example, the square root of 144 is 12, because 12 × 12 = 144. Every positive number actually has two square roots, one positive and one negative (12 and −12 both square to 144), but “the square root” by convention refers to the positive one — called the principal root. The calculator always returns this principal root.

Cube roots and other nth roots

The nth root generalizes this idea: the nth root of n is the value that, raised to the power n, gives n back. Setting the root degree to 3 gives you a cube root: the cube root of 27 is 3, because 3 × 3 × 3 = 27. Cube roots behave differently from square roots on negative numbers — see below. Degrees higher than 3 (4th root, 5th root…) follow the same pattern; they’re just used less often outside science and engineering.

Simplifying a root into radical form

A root is in simplified radical form when nothing more can be pulled out from under the root sign. The calculator does this by breaking the radicand into prime factors and pulling out any group of n matching primes (for a degree-n root), where n is the root degree. Take √72: 72 = 2³ × 3². Two of those 3s pair up and come out as a plain 3; two of the 2s pair up and come out as 2; one lone 2 stays behind. That leaves 6√2 — 6 outside the root, 2 underneath. If nothing pairs up at all (like with √2, where 2 has no repeated factors), the root is already in simplest form and the calculator won’t show a separate radical line.

Negative numbers under a root

Whether a negative radicand has a real answer depends entirely on whether the root degree is even or odd. An odd degree root of a negative number is real and negative: the cube root of −27 is −3, because −3 × −3 × −3 = −27. An even degree root of a negative number — like the square root of −4 — has no real-number answer, because no real number multiplied by itself an even number of times can produce a negative result. The calculator flags this case instead of guessing.

Roots and fractional exponents

Every root can be rewritten as a power with a fractional exponent: the nth root of n is the same as n raised to the power 1/n. So √x = x^(1/2), and the cube root of x = x^(1/3). This is exactly how the calculator computes the decimal value internally, then double-checks it by raising the result back to the power you entered (that’s the “power check” you see under the answer).

Rounding and decimal precision

Most roots don’t come out to a clean decimal — √2 = 1.4142135623730951… goes on forever without repeating. The calculator rounds this to a fixed number of decimal places, 6 by default, which you can change from 0 to 10 in Advanced options. A lower setting is handy when you just need a quick estimate; a higher one when precision matters, for example in engineering calculations.

Step-by-step examples

Example 1: an exact square root — √144

144 factors as 2⁴ × 3². Pairing up the factors for a degree-2 root: 2² × 3 = 12, with nothing left under the root. So √144 = 12 exactly, and the check confirms it: 12² = 144.

Example 2: simplifying an irrational square root — √72

72 factors as 2³ × 3². One pair of 2s and one pair of 3s come out; one lone 2 stays under the root. That gives 6√2, which as a decimal (6 places) is √72 ≈ 8.485281.

You can reach the same decimal by hand with the averaging method: start with a guess, say 8.5, then repeatedly replace it with (guess + 72/guess) / 2. First step: (8.5 + 72/8.5) / 2 ≈ 8.4853 — already within 0.001 of the exact value after a single step.

Example 3: an already-simplified square root — √2

2 has no repeated prime factors, so there’s nothing to pull out — the radical is already in simplest form (coefficient 1). As a decimal, √2 ≈ 1.414214.

Example 4: cube roots of positive and negative numbers — ∛27 and ∛−27

27 = 3³, so the cube root of 27 is exactly 3 (check: 3³ = 27). Flip the sign of the radicand and the sign of the result flips too, because the root degree is odd: the cube root of −27 is exactly −3 (check: (−3)³ = −27).

Example 5: simplifying a cube root — ∛16

16 = 2⁴. For a degree-3 root, only complete groups of three matching factors come out: one group of three 2s becomes a plain 2, leaving one 2 behind. That gives 2∛2, or ≈ 2.519842 as a decimal.

Example 6: a fractional radicand and a negative even root

√0.25 = 0.5 exactly (0.5 × 0.5 = 0.25) — the calculator skips the radical-form step here, since factoring only applies to whole-number radicands. By contrast, √−4 has no real result: −4 is negative and the root degree (2) is even, so the calculator reports that instead of a number.

Frequently asked questions

How do you calculate a square root by hand, without a calculator?

Bracket the number between two perfect squares, then refine with the averaging method: replace your guess x with (x + n/x) / 2 and repeat. For √72, starting from 8.5 gives (8.5 + 72/8.5) / 2 ≈ 8.4853 after one step — already accurate to three decimal places, and further repetitions get you closer still.

Can you take the square root of a negative number?

Not as a real number. A square root asks “what number times itself gives this result?” — and any real number multiplied by itself, positive or negative, always produces a non-negative result. So no real number squares to a negative value, which is why √−4 has no real answer. (Negative numbers do have square roots among the complex numbers, using the imaginary unit i, but that’s a separate number system.)

How do you simplify a square root into radical form, like √72 = 6√2?

Break the number into prime factors and pull out any pair of matching primes as a single factor outside the root. For 72 = 2³ × 3², one pair of 2s and one pair of 3s come out (giving 2 × 3 = 6), leaving one unpaired 2 under the root — so √72 = 6√2. If no prime factor repeats, the root is already fully simplified.

What is an nth root, and how does it differ from a square root?

An nth root of a number n is the value that, raised to the power n, gives n back — the square root is just the special case where the degree is 2. A cube root (degree 3) or a fifth root (degree 5) work the same way, just with three or five matching factors, instead of two, needed to pull a number out from under the root.

How do you calculate a cube root?

Find the value that, multiplied by itself three times, gives the original number. For a whole number, factor it into primes and pull out complete groups of three matching factors: 27 = 3³ gives exactly 3, while 16 = 2⁴ only has one complete group of three 2s, leaving 2∛2 ≈ 2.519842. Cube roots of negative numbers are real: the cube root of −27 is −3.

What’s the relationship between a root and a fractional exponent?

An nth root is the same operation as raising a number to the power 1/n: the square root of x equals x^(1/2), and the cube root of x equals x^(1/3). This is why calculators (and this tool internally) compute roots using the power function, then verify the result by raising it back to the nth power.

Can a square root be negative?

A positive number always has two square roots, one positive and one negative — 12 and −12 both square to 144. By convention, “the square root,” written √144, refers only to the non-negative one (the principal root); the negative root is written −√144 when it’s needed, for example when solving an equation like x² = 144.

How do you find an nth root on a basic calculator without a root button?

Use the power function: raise the number to 1 divided by the degree. For a cube root of 16, that’s 16^(1/3), which most calculators accept as 16 ^ ( 1 ÷ 3 ). This gives the same result as a dedicated root key, since an nth root and the power 1/n are mathematically identical.

How many decimal places should a root be rounded to?

It depends on what you need it for: 2 to 4 decimal places is plenty for everyday estimates, while engineering or scientific work often calls for 6 or more. This calculator defaults to 6 decimal places and lets you set anywhere from 0 to 10 in the advanced options, so you can match the precision to the task.

How do you tell if a number is a perfect square or a perfect cube?

Factor it into primes: it’s a perfect square if every prime factor’s exponent is even (144 = 2⁴ × 3² — both exponents even), and a perfect cube if every exponent is a multiple of three (27 = 3³). Equivalently, run it through this calculator — a perfect square or cube comes back as an exact whole-number result with no leftover radical.

Sources