Square Root / Nth Root

Finding the nth Root of a Number

The nth root of a number is the value that, when raised to the nth power, gives back the original number. Enter a number and a root degree (2 for a square root, 3 for a cube root, and so on), and this calculator finds that value.

A square root (degree 2) and a cube root (degree 3) are the two most commonly used cases, but the same idea extends to any whole-number degree. A negative number only has a real nth root when the degree is odd — an even root (like an ordinary square root) of a negative number isn’t a real number at all, since no real number multiplied by itself an even number of times can ever produce a negative result.

The Formula

$$\sqrt[\vB{n}]{\vA{x}} = \vA{x}^{\frac{1}{\vB{n}}}$$

where $\vA{x}$ is the number and $\vB{n}$ is the root degree. Checking the answer is straightforward: raising the result back to the $\vB{n}$th power should reproduce $\vA{x}$.

Worked Example

1. Square root: $\sqrt{\vA{144}} = 12$, since $12^2 = 144$.
2. Cube root: $\sqrt[3]{\vA{27}} = 3$, since $3^3 = 27$.
3. Negative number, odd degree: a real answer still exists — $\sqrt[3]{\vA{-8}} = -2$, since $(-2)^3 = -8$.
4. Negative number, even degree: an even root of a negative number, like $\sqrt{-8}$, has no real answer — this calculator flags that case rather than returning a misleading result.

Key Factors to Consider

• A positive number technically has two real square roots, though only the positive one is usually meant by default. Both 12 and -12 square to 144, but the ”√” symbol conventionally refers to the positive (principal) root — this calculator, like standard mathematical convention, returns the principal root.
• Roots and exponents are exact inverses of each other, which is why the fractional-exponent formula works. Taking the nth root of a number is mathematically identical to raising it to the power of 1/n — this is exactly the relationship the Exponent Calculator uses in the opposite direction, and understanding one helps make sense of the other.
• Even-degree roots of negative numbers do exist in the complex number system, just not in the real numbers this calculator computes. A negative number raised to an even root produces an “imaginary” result mathematically — this calculator is deliberately scoped to real-number answers, matching what’s actually meaningful for the vast majority of everyday and practical root calculations.
• Higher-degree roots grow much more slowly than the underlying number does. As a number gets larger, its square root grows slower than the number itself, its cube root grows even more slowly, and so on for higher degrees — this diminishing-growth property is why roots are commonly used to “compress” a wide range of values into a more manageable scale in statistics and data analysis.

Common Mistakes

• Expecting a real answer from an even-degree root of a negative number. A square root, fourth root, or any other even-degree root of a negative number has no real solution — this calculator flags that case rather than silently returning an incorrect number.
• Forgetting that the principal square root is only the positive one. Both 12 and -12 square to 144, but “√144” conventionally means just the positive root — don’t assume the negative root is automatically included.
• Confusing a root degree with the number being rooted. Entering the degree and the number in the wrong fields produces a completely different (and usually much larger or smaller) answer — double-check which field is which before reading the result.

Useful to Know

• Need to go the opposite direction — raising a number to a power instead of finding its root? Exponent Calculator handles that inverse operation.
• Working with logarithms alongside roots and exponents? Logarithm Calculator is the third piece of that same mathematical relationship.
• Need the full scientific calculator for a more complex expression? Scientific Calculator supports roots, exponents, and much more in one expression.