Root Calculator
Use this calculator to find the root of any number. You can calculate the square root (√), cube root (∛), or the nth root for any positive number.
Your Result
Select a root type, enter a number, and click "Calculate".
What is a Root in Mathematics?
In mathematics, finding the root of a number is the inverse operation of raising a number to a power. For a number x, its nth root is a number y such that when y is multiplied by itself n times, the result is x. This relationship is expressed with a radical symbol (√).
n√ x = y (because yn = x)
- Radicand (x): The number under the radical symbol whose root is being calculated.
- Index (n): The degree of the root. For a square root, the index is 2 and is usually omitted.
- Radical (√): The symbol indicating that a root should be taken.
Types of Roots
Our calculator can handle the most common types of roots:
- Square Root (n=2): The most common root. The square root of 9 is 3 because 32 = 9.
- Cube Root (n=3): The number that produces the radicand when cubed. The cube root of -27 is -3 because (-3)3 = -27.
- Nth Root: A generalized root where 'n' can be any number. For example, the 4th root of 16 is 2 because 24 = 16.
How the Root Is Calculated
Finding the nth root of a number x is equivalent to raising x to the power of 1/n.
n√ x = x(1/n)
Our calculator uses this formula. However, a special consideration is needed for even roots (like square roots) of negative numbers, which result in imaginary numbers. This calculator focuses on real number results and will show an error in such cases.
Real-World Applications of Roots
Roots are fundamental in many fields:
- Geometry: Using the Pythagorean theorem (a² + b² = c²) to find the length of a side of a right triangle involves taking a square root.
- Physics: Calculating distance, velocity, or time in various kinematic equations.
- Finance: Finding the average rate of return on an investment over multiple years requires calculating a geometric mean, which uses nth roots. Our Financial Calculators may help with related problems.
- Statistics: Calculating standard deviation involves a square root.
Related Math Calculators
For more mathematical tools, check out our other calculators:
- Math Calculators - Our main hub for various mathematical utilities.
- Ratio Calculator - Solve proportions and simplify ratios.
- Exponent Calculator (Coming Soon) - The inverse of this calculator, used for raising numbers to a power.