Square root of a perfect square
Use index 2 to find the principal square root of a positive number.
√64 = 64^(1/2)
Result: 8
Root Calculator
Use this free online root calculator to find the square root, cube root, or any nth root of a number. Get step-by-step solutions and explanations.
Root Calculator
Enter a number and the root index (2 for square root, 3 for cube root, etc.), then click Calculate to find the result with step-by-step solution.
Result
What is a Root?
In mathematics, a root of a number x is another number, which when raised to a given power n, equals x. In other words, the nth root of x is a number r such that r^n = x.
For example, the square root (2nd root) of 9 is 3, because 3^2 = 9. The cube root (3rd root) of 8 is 2, because 2^3 = 8.
Types of Roots
- Square Root (2nd Root): The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol √.
- Cube Root (3rd Root): The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted by the symbol ∛.
- Fourth Root: The fourth root of a number is a value that, when raised to the power of 4, gives the original number. It is denoted by ∜.
- nth Root: The nth root of a number is a value that, when raised to the power of n, gives the original number. It is denoted by the radical symbol with a small n: ⁿ√.
Root Calculation Formula
The formula for calculating the nth root of a number x is:
ⁿ√x = x^(1/n)
Where n is the root index (2 for square root, 3 for cube root, etc.) and x is the number for which we want to find the root.
How to Use the Root Calculator
- Enter the number for which you want to find the root.
- Enter the root index (2 for square root, 3 for cube root, etc.).
- Click 'Calculate' to find the root.
- View the result and step-by-step solution.
Use index 2 for square roots, 3 for cube roots, and any positive odd index when working with negative numbers.
Applications of Roots in Real Life
- Engineering and construction for calculating dimensions and structural stability.
- Physics for determining relationships between variables in equations.
- Finance for calculating compound interest and investment growth.
- Computer graphics for scaling and transforming objects in 3D space.
Root Calculator Examples
Cube root of a negative number
Odd roots can be evaluated for negative values without leaving the real number system.
∛(-125) = (-125)^(1/3)
Result: -5
Fourth root for repeated powers
Higher-order roots are useful when reversing exponential growth or area/volume scaling.
⁴√81 = 81^(1/4)
Result: 3
Root Calculator FAQ
What is the difference between a square root and an nth root?
A square root is specifically the second root. An nth root generalizes that idea and asks which value raised to the power n returns the original number.
Why can't I calculate an even root of a negative number?
In the real number system, even powers of real numbers are never negative. That means values like the square root of -16 do not have a real-number result.
Can this calculator handle cube roots of negative numbers?
Yes. Odd roots of negative numbers are valid real numbers, so the calculator can return results like ∛(-27) = -3.
When should I use a root calculator instead of an exponent calculator?
Use a root calculator when you need to reverse a power operation, such as finding the side length from an area or solving x^n = a for x.