Root Calculator – Calculate Square, Cube and Nth Roots
Calculate any nth root instantly with our free Root Calculator. Find square roots, cube roots, or any higher-order roots with full precision. Enter your radicand and root index to get results — supports negative numbers for odd roots.
How to Use This Root Calculator
Enter the radicand
Type the number you want to find the root of. This can be any positive number, or negative for odd roots.
Set the root index
Enter 2 for square root, 3 for cube root, or any other positive integer for higher-order roots.
Calculate instantly
Click Calculate to see the result. The calculator shows the nth root with full precision.
Common Root Values Reference
| Number | Square Root (√) | Cube Root (³√) | 4th Root (⁴√) |
|---|---|---|---|
| 1 | 1.000 | 1.000 | 1.000 |
| 4 | 2.000 | 1.587 | 1.414 |
| 8 | 2.828 | 2.000 | 1.682 |
| 9 | 3.000 | 2.080 | 1.732 |
| 16 | 4.000 | 2.000 | 2.000 |
| 25 | 5.000 | 2.924 | 2.236 |
| 27 | 5.196 | 3.000 | 2.280 |
| 64 | 8.000 | 4.000 | 2.828 |
| 81 | 9.000 | 4.327 | 3.000 |
| 100 | 10.000 | 4.642 | 3.162 |
Note: Values rounded to 3 decimal places. Perfect squares and cubes have integer roots.
Understanding nth Roots
The nth root of a number x is the value that, when raised to the power of n, equals x. In mathematical notation: if yⁿ = x, then y = ⁿ√x. The square root (n=2) and cube root (n=3) are the most common, but you can calculate any positive integer root.
Square roots appear everywhere in math and science. The Pythagorean theorem uses them to find distances. Standard deviation in statistics is a square root. Even the quadratic formula has a square root in it. Cube roots show up in volume calculations — if you know a cube's volume, the cube root gives you the side length.
Higher roots have practical uses too. The 4th root appears in physics formulas for radiation and in engineering stress calculations. Financial analysts use roots when calculating compound annual growth rates over multiple periods.
Key insight: Taking the nth root is the same as raising to the power of 1/n. So ³√8 = 8^(1/3) = 2. This relationship lets calculators compute roots using exponentiation.
Root Calculation Tips and Rules
Even roots of negative numbers
Square roots, 4th roots, and all even roots of negative numbers produce complex (imaginary) results. There's no real number that squares to -4. This calculator shows "Undefined" for these cases since it works with real numbers only.
Odd roots preserve sign
Cube roots, 5th roots, and all odd roots of negative numbers are negative. The cube root of -8 is -2 because (-2)³ = -8. This calculator handles negative radicands for odd root indices.
Perfect powers have integer roots
When a number is a perfect square (4, 9, 16, 25...), its square root is a whole number. Perfect cubes (8, 27, 64, 125...) have integer cube roots. Recognizing these speeds up mental math.
Root of 1 is always 1
No matter what root index you use, the nth root of 1 equals 1. This is because 1 raised to any power stays 1. Similarly, the nth root of 0 is always 0.
Frequently Asked Questions
What is the square root of 2?
√2 ≈ 1.41421356... It's an irrational number, meaning the decimal never ends or repeats. This value appears in geometry — it's the diagonal of a unit square. The ancient Greeks discovered √2 cannot be written as a fraction.
Can I calculate the 5th root of a number?
Yes. Enter your number as the radicand and 5 as the root index. The 5th root of 32 is 2 because 2⁵ = 32. Higher roots get smaller — the 10th root of 1024 is also 2.
Why is the cube root of a negative number defined?
Odd roots work with negative numbers because a negative times itself an odd number of times stays negative. (-3)³ = -27, so ³√(-27) = -3. Even roots like square root have no real solution for negatives.
What's the difference between square root and principal square root?
Every positive number has two square roots: one positive, one negative. Both 3 and -3 square to 9. The "principal" square root is the positive one — that's what the √ symbol means. This calculator returns the principal root.
How accurate are the results?
Results use JavaScript's floating-point precision (about 15-17 significant digits). For most practical purposes this is more than enough. Extremely large or small numbers may have minor rounding in the last digits.