Root Calculator

What a root actually asks

A root undoes a power. The square root of a number is whatever, multiplied by itself once, gets back to that number; the cube root is whatever, multiplied by itself twice more, gets back to it; and an nth root generalises the idea to n copies of the same factor multiplied together. Written as a radical, √48 asks "what, squared, makes 48?" and ∭54 asks "what, cubed, makes 54?"

Most numbers do not have a tidy whole-number answer to that question. 48 is not anybody squared, so its square root is an irrational number, one whose decimal digits never settle into a repeating pattern and never stop. What this calculator does differently is refuse to just round that number off: it factors 48 into its prime building blocks first, pulls out every pair it can, and gives you the exact leftover radical, 4√3, alongside the decimal.

Simplifying a radical by prime factorisation

Every whole number breaks down into a unique set of prime factors, and that breakdown is the key to simplifying a root. For a square root, any prime that appears twice in the factorisation can be pulled out as a single copy of itself; for a cube root, a prime needs to appear three times before a copy comes out; for an nth root generally, a prime needs n copies before one comes out, and whatever is left over (fewer than n copies) stays trapped inside the radical.

Take 48. It factors into 2 × 2 × 2 × 2 × 3, or 2⁴ × 3. The four 2s split into two matching pairs, and each pair steps outside the square root as a single 2, so two pairs become 2 × 2 = 4 outside the root. The lone 3 has no partner, so it stays inside.

Cube roots work the same way but need triples instead of pairs. 54 factors into 2 × 3³. The three 3s form one complete triple, which steps out of the cube root as a single 3, leaving the 2 behind: ∭54 = 3∭2. A fourth root needs quadruples: 1250 factors into 2 × 5⁴, and the four 5s form one complete group of four, stepping out as a single 5 and leaving the 2 behind, so the 4th root of 1250 is 5·⁴√2. When no prime in the factorisation reaches the needed count, as with a prime number like 97, nothing can step outside and the radical is already in its simplest form.

Negative whole numbers follow the same factorisation, applied to the number's size, with the sign settled separately: an odd root of a negative number is a real, negative answer (∭−27 = −3, since −3 × −3 × −3 = −27), while an even root of a negative number has no real answer at all, covered below.

Perfect squares, cubes and other exact powers

A perfect square is what you get from squaring a whole number: 49 is a perfect square because 7 × 7 = 49, and its square root is exactly 7, not a radical at all. The same idea extends to any power: 1000 is a perfect cube (10 × 10 × 10), and 1024 is a perfect 10th power (2 multiplied by itself ten times). This calculator checks for that exactly, using whole-number arithmetic rather than a rounded decimal guess, so a very slightly-off answer like 6.99999999 is never mistaken for a perfect square that should have been 7.

A quick reference for the smaller perfect powers:

Roots of negative numbers

Whether a negative number has a real root depends entirely on whether the root is odd or even. An odd number of negative factors multiplied together stays negative, so −3 × −3 × −3 = −27 and the cube root of −27 is a perfectly ordinary −3. An even number of negative factors always cancels back to positive, so there is no real number that squares to a negative value: nothing, multiplied by itself, produces a negative result, whether that nothing was positive, negative, or in between. That is why this calculator answers √(−4) with a plain explanation rather than a number: it is not that the sum is hard, it is that no real answer exists. (A second family of numbers, the complex numbers, does handle this case, but they sit outside what this tool covers.)

Estimating a square root by hand

Before reaching for a calculator, a square root can be pinned down closely with two short steps, a method sometimes called the Babylonian or Heron's method. First, bracket the number between two perfect squares to get a starting whole-number guess; second, average that guess with the number divided by the guess, which pulls the estimate much closer to the true value.

Take 40. It sits between 36 (6²) and 49 (7²), so the square root of 40 is somewhere between 6 and 7, and 6 is the closer whole number to start from. Dividing 40 by 6 gives 6.667, and averaging 6 with 6.667 gives 6.333. The true value is 6.325, so one averaging step already lands within 0.01 of the answer. Repeating the process, this time averaging 6.333 with 40 ÷ 6.333 (which is 6.316), gives 6.3246, matching the true value to four decimal places.

The same two steps work for any square root: bracket it between consecutive perfect squares from the reference table above, then average and repeat until the guess stops moving much.

Reading exact, simplified and decimal results

Three different things can appear in the answer here, and they mean different things. An exact whole number, such as 7 for √49, means the input was a perfect power outright. A simplified radical, such as 4√3, is also an exact answer: it is not an approximation of anything, it is the precise value of √48 written with the smallest possible whole number outside the root sign. Only the decimal line underneath, 6.928203230…, is an approximation, truncated rather than rounded and marked with an ellipsis to show it carries on forever without repeating. A decimal input such as 6.25 skips the radical step entirely, since simplifying into whole-number factors only makes sense for whole numbers, and returns a precise decimal answer instead.

Questions people ask

Perfect-power checks and prime factorisation here run on exact integer arithmetic throughout, never on a rounded decimal guess. For working with fractions rather than roots, try the fraction calculator; for solving a quadratic equation, where square roots also turn up, the quadratic formula calculator is on our build list.