Calculate the square root, cube root, or any nth root of a number.
A root answers the reverse of an exponent question: "what number, multiplied by itself n times, gives me this result?" The square root of 64 is 8 because 8 × 8 = 64. The cube root of 64 is 4 because 4 × 4 × 4 = 64.
Square roots show up constantly in geometry (like finding a side length from an area) and statistics (like standard deviation). Cube roots come up when reversing volume calculations. Higher-degree roots are less common day-to-day but appear in engineering and finance formulas involving compound rates.
Enter the number you want to find the root of, then enter the degree of the root — 2 for square root, 3 for cube root, 4 for fourth root, and so on. Click Calculate, and the tool returns the result. For even-degree roots of negative numbers, the calculator will show an error, since these don't have a real-number solution — more on that below.
Roots and exponents are inverse operations, similar to how subtraction reverses addition or division reverses multiplication. If 5² = 25, then the square root of 25 reverses that operation to return 5. If 3³ = 27, then the cube root of 27 returns 3. This relationship is why roots can also be written as fractional exponents: the nth root of x is the same as x^(1/n). Understanding this connection makes it easier to see why root rules follow directly from exponent rules — they're really the same mathematical operation viewed from opposite directions.
An even-degree root (square root, fourth root, sixth root, etc.) of a negative number has no solution among real numbers, because no real number multiplied by itself an even number of times can produce a negative result — a negative number squared is always positive, so there's no real number that squares to give a negative. Odd-degree roots (cube root, fifth root, etc.) work differently: a negative number multiplied by itself an odd number of times stays negative, so odd roots of negative numbers do have valid real solutions — for example, the cube root of −27 is −3, since (−3) × (−3) × (−3) = −27.
A perfect square is a number that's the result of squaring a whole number — 1, 4, 9, 16, 25, and so on. A perfect cube works the same way for cubing — 1, 8, 27, 64, 125. Recognizing these patterns makes mental estimation easier: since 49 and 64 are perfect squares (7² and 8²), you can quickly tell that the square root of 55 falls somewhere between 7 and 8, without needing a calculator for a rough estimate. This kind of estimation skill is genuinely useful for sanity-checking a calculator's output or working through problems where an exact decimal isn't necessary.
Roots appear across many practical and technical fields. In geometry, the Pythagorean theorem uses a square root to find the length of a triangle's hypotenuse. In statistics, standard deviation — a measure of how spread out a data set is — is calculated as the square root of variance. In finance, calculating the annualized return of a multi-year investment often involves taking an nth root to reverse compound growth over time. In physics, root calculations show up in formulas involving energy, motion, and wave behavior, since many physical relationships involve squared or cubed quantities that need to be reversed to solve for an underlying variable.
Before relying on a calculator, it's worth knowing how to simplify a square root manually for numbers that aren't perfect squares. The trick is to factor out the largest perfect square divisor: the square root of 72 can be broken down as the square root of 36 times 2, and since 36 is a perfect square (6²), this simplifies to 6 times the square root of 2. This gives a cleaner, exact form (6√2) rather than a rounded decimal, which is often preferred in algebra and geometry contexts where an exact value matters more than a decimal approximation.
What's the difference between a square root and a cube root? A square root reverses squaring (raising to the power of 2), while a cube root reverses cubing (raising to the power of 3). Square roots only have real solutions for non-negative numbers, while cube roots have real solutions for any number, including negatives.
Why does my calculator show an error for the square root of a negative number? Because there's no real number that, when squared, produces a negative result. The square root of a negative number exists only in the realm of imaginary numbers, which this calculator doesn't compute.
Can I calculate roots with decimal degrees, like a 2.5 root? This calculator is built around whole-number root degrees (2, 3, 4, etc.), which cover the vast majority of practical use cases. Fractional-degree roots are mathematically valid but rarely needed outside specialized contexts.
How is a root different from a fractional exponent calculation? They're mathematically identical — the nth root of x equals x raised to the power of 1/n. This calculator computes roots directly, but you'd get the same result using a scientific calculator's exponent function with a fractional power.
Is there a shortcut for cube roots the way there is for square roots? Recognizing perfect cubes (1, 8, 27, 64, 125) helps with quick estimation the same way perfect squares do, though the pattern is less commonly memorized.
Why do some calculators show a slightly different decimal for the same root? Small differences usually come down to how many decimal places are displayed or rounded — the underlying mathematical value is the same.
Can roots be negative themselves, as opposed to the number under the root being negative? Every positive real number technically has two square roots, one positive and one negative, though the standard root calculator convention returns only the positive (principal) root.
Last reviewed by Mehmed on July 11, 2026.