Square Root and Nth Root Calculator
Calculate square, cube, and nth roots, simplify exact integer radicals, and check the result by raising it to the selected index.
Square root and nth root formulas
The nth root of a number a is a value r such that rn = a. It is written ⁿ√a or a1/n. A square root uses n=2, and a cube root uses n=3.
For a positive number and an even index, the radical symbol gives the nonnegative principal root. The equation xn=a has two real solutions, ±ⁿ√a. With an odd index, a real number has one real nth root, including when the number is negative.
To simplify an integer radical, factor the radicand and move every complete group of n equal prime factors outside the radical.
Checked examples
| Input | Index | Simplified form | Decimal root |
|---|---|---|---|
| 9 | 2 | 3 | 3 |
| 2 | 2 | √2 | ≈1.41421356237 |
| 12 | 2 | 2√3 | ≈3.46410161514 |
| 0 | 2 | 0 | 0 |
| 0.25 | 2 | Decimal input | 0.5 |
| −8 | 3 | −2 | −2 |
| 81 | 4 | 3 | 3 |
| 32 | 5 | 2 | 2 |
| 54 | 3 | 3∛2 | ≈3.77976314968 |
| 10,000,000,000 | 2 | 100000 | 100000 |
How to use the calculator
- Enter the number whose root you need.
- Enter 2 for square root, 3 for cube root, or another whole-number index through 100.
- Select Calculate root.
- Review the decimal value, simplified radical, real-solution set, and power check.
Limits and interpretation
- The number must be finite and no greater than 1e100 in magnitude.
- The root index must be a whole number from 2 to 100.
- A negative radicand with an even index has no real root; this real-number calculator reports that condition instead of inventing a real result.
- Exact radical simplification is available for integer magnitudes up to one trillion.
- Non-integer inputs receive a decimal root but are not rewritten as an exact radical.
- Decimal results use double-precision arithmetic and are displayed to approximately 14 significant digits.
Frequently asked questions
Why does √9 equal 3 rather than ±3?
The radical symbol denotes the principal, nonnegative square root. The equation x²=9 has two solutions, −3 and 3, both shown in the real-solutions field.
Can the calculator simplify radicals?
Yes, for integer radicands up to one trillion. For example, √12 becomes 2√3 and ∛54 becomes 3∛2.
Can an odd root use a negative number?
Yes. An odd power preserves the sign, so ∛(−8)=−2 and the fifth root of −32 is −2.
What is a perfect square or perfect nth power?
It is an integer whose selected root is also an integer. Examples include 81 as a perfect square and 32 as a perfect fifth power.
Related calculators
Use the Quadratic Formula Calculator for equation roots and discriminants, the Scientific Notation Calculator for very large or small values, the Logarithm Calculator for logarithms and inverse powers, or the Fraction Calculator for exact rational arithmetic.