Use this free online root calculator to find the nth root of any number, including square roots, cube roots, and higher-order roots. Simply enter a number and the root degree to get an instant, precise result without doing the math by hand.
How to Use the Root Calculator
Enter the number you want to find the root of in the "Number" field.
Enter the root degree in the "Root Degree (n)" field — use 2 for a square root, 3 for a cube root, or any other whole number.
Click "Calculate" to instantly see the result.
If the inputs are invalid, such as an even root of a negative number, an error message will explain why.
What an nth Root Means
The nth root of x is the number that gives x when multiplied by itself n times. The cube root of 1000 is 10 because 10 × 10 × 10 = 1000. Taking a root is the same as raising to the power 1/n, which is exactly how the calculator computes it.
For negative numbers with an odd degree, the tool takes the root of the positive value and restores the minus sign, so the cube root of −8 correctly comes out as −2. Results are rounded to 10 decimal places, which hides the tiny floating-point error that would otherwise turn the cube root of 1000 into 9.999999999999998.
Useful Roots to Know
Number
Square root
Cube root
2
1.4142135624
1.2599210499
3
1.7320508076
1.4422495703
5
2.2360679775
1.7099759467
10
3.1622776602
2.15443469
To estimate a root without a calculator, find the nearest perfect powers on either side. 50 lies between 49 (7²) and 64 (8²) and is much closer to 49, so its square root is a little above 7. The calculator gives 7.0710678119.
Practical Uses
Annual growth rate (CAGR): if an investment grew from ₹1,00,000 to ₹2,00,000 in 10 years, the growth multiple is 2. The 10th root of 2 is 1.0717734625, so it grew at about 7.18% a year on average.
Side from area: a square plot of 1,200 sq ft has sides of √1200 = 34.641 ft.
Side from volume: a cube-shaped tank that holds 2 cubic metres (2,000 litres) has an inside edge of the cube root of 2, about 1.26 m.
Music: on a standard keyboard each semitone raises the frequency by the 12th root of 2, about 1.0595.
Things to Note
Every positive number has two square roots, one positive and one negative (both 5 and −5 square to 25). The calculator, like the √ symbol, returns the positive one.
Irrational roots such as √2 never end or repeat, so the 10-decimal answer is an approximation, though far more precise than everyday work needs.
Stick to whole-number degrees. A negative degree gives the reciprocal (degree −2 on 16 gives 0.25), and fractional degrees act as powers, which is rarely what you want from a root calculator.
Frequently Asked Questions
It raises the number (radicand) to the power of 1 divided by the root degree (n), i.e. result = sign(radicand) × |radicand|^(1/n). For example, the 3rd root of 27 is calculated as 27^(1/3) = 3.
Yes, but only for odd root degrees. For an even root degree (like 2 or 4) of a negative number, the result is not a real number, so the calculator shows "Not a real number for an even root of a negative number" instead of a value.
Split the number into a perfect square times what is left: 72 = 36 × 2, so √72 = 6√2. The calculator gives the decimal value, 8.4852813742, which is 6 times the square root of 2.
Divide the final value by the starting value, take the root whose degree is the number of years, and subtract 1. Money that tripled in 15 years grew at the 15th root of 3 minus 1, about 7.6% a year.
It is larger than the number itself. For example, √0.25 = 0.5 and the cube root of 0.001 is 0.1, because multiplying a fraction by itself makes it smaller.