Find The Nth Root Calculator

Find the Nth Root

The nth root of a number ‘a’ (radicand) is a number ‘x’ such that xⁿ = a. We calculate it as x = a^(1/n).

Index (n)	Root of Radicand
Enter values to see results

Table showing roots for indices around the input value ‘n’ for the given radicand.

Chart showing how the root (y-axis) changes with the index (x-axis) for the given radicand, and the calculated nth root.

What is the Nth Root Calculator?

An Nth Root Calculator is a tool used to find the ‘nth’ root of a given number, which is also known as the radicand. If you have a number ‘a’ and you want to find its nth root, you are looking for a number ‘x’ such that when ‘x’ is multiplied by itself ‘n’ times, the result is ‘a’ (i.e., xⁿ = a). The ‘n’ is called the index of the root.

For example, the 2nd root is the square root, and the 3rd root is the cube root. This calculator allows you to find any root (2nd, 3rd, 4th, 5th, etc.) of a number. It’s useful in various fields like mathematics, engineering, finance (for compound interest over fractional periods, although that’s more complex), and science.

Anyone who needs to solve equations involving powers and roots, or wants to reverse an exponentiation operation, can use the Nth Root Calculator. A common misconception is that roots are always smaller than the original number; while true for numbers greater than 1, for numbers between 0 and 1, the nth root is larger than the number itself.

Nth Root Formula and Mathematical Explanation

The nth root of a number ‘a’ is a number ‘x’ that, when raised to the power of ‘n’, equals ‘a’. The formula is:

x = ⁿ√a = a^1/n

Where:

• ‘a’ is the radicand (the number you are finding the root of).
• ‘n’ is the index (the root you are looking for, e.g., 2 for square root, 3 for cube root).
• ‘x’ is the nth root.

The calculation involves raising the radicand ‘a’ to the power of ‘1/n’.

Variables Table:

Variable	Meaning	Unit	Typical Range
a (Radicand)	The number whose root is being calculated	Dimensionless (or units depend on context)	Non-negative if ‘n’ is even; any real number if ‘n’ is odd
n (Index)	The degree of the root	Dimensionless	Integer > 1 (or real number > 0 for generalized roots)
x (Root)	The result of the nth root calculation	Same as ‘a’ if ‘a’ has units	Real number

Practical Examples (Real-World Use Cases)

Let’s look at a couple of examples using the Nth Root Calculator:

Example 1: Finding the 4th root of 81

• Radicand (a) = 81
• Index (n) = 4
• We are looking for a number x such that x⁴ = 81.
• Using the formula: x = 81^(1/4) = 3.
• So, the 4th root of 81 is 3 (because 3 * 3 * 3 * 3 = 81).

Example 2: Finding the 5th root of 32

• Radicand (a) = 32
• Index (n) = 5
• We need x such that x⁵ = 32.
• Using the formula: x = 32^(1/5) = 2.
• The 5th root of 32 is 2 (because 2 * 2 * 2 * 2 * 2 = 32).

Example 3: Geometric Mean

The geometric mean of ‘n’ numbers is the nth root of their product. If you have growth rates of 1.10, 1.05, and 1.12 over three periods, the average growth rate is the cube root of (1.10 * 1.05 * 1.12), which is (1.2936)^1/3 ≈ 1.0898. The Nth Root Calculator can find this.

How to Use This Nth Root Calculator

1. Enter the Radicand (a): Input the number for which you want to find the root into the “Number (Radicand, a)” field.
2. Enter the Index (n): Input the root you want to find (e.g., 2 for square root, 3 for cube root, etc.) into the “Root (Index, n)” field.
3. View the Results: The calculator automatically displays the primary result (the nth root) and intermediate values as you type.
4. Interpret the Table and Chart: The table shows roots for indices near your input ‘n’, and the chart visualizes how the root changes with the index for the given radicand, highlighting your calculated point.
5. Reset or Copy: Use the “Reset” button to clear inputs and results, or “Copy Results” to copy the main result and inputs.

Understanding the result is straightforward: the “Primary Result” is the number that, when raised to the power of the index ‘n’, equals your radicand ‘a’. Our Square Root Calculator is a specialized version for n=2.

Key Factors That Affect Nth Root Results

The result of an nth root calculation is directly determined by two factors:

1. The Radicand (a): The number you are finding the root of. As the radicand increases (for a fixed positive index > 1), the nth root also increases. If the radicand is between 0 and 1, the nth root will be larger than the radicand.
2. The Index (n): The degree of the root. For a radicand greater than 1, as the index ‘n’ increases, the nth root decreases and approaches 1. For a radicand between 0 and 1, as ‘n’ increases, the nth root increases and approaches 1. The index must be a positive number, usually an integer greater than 1, but generalized roots can have non-integer indices. Our Cube Root Calculator handles the case n=3.
3. Sign of the Radicand: If the index ‘n’ is even, the radicand ‘a’ must be non-negative to yield a real number root. If ‘n’ is odd, the radicand ‘a’ can be any real number, and the root will have the same sign as ‘a’.
4. Magnitude of the Radicand: Very large or very small radicands can result in very large or very small roots, respectively, especially for small indices.
5. Nature of the Index: While usually an integer, ‘n’ can be a rational number, leading to more complex interpretations involving powers and roots.
6. Computational Precision: Calculators use algorithms that have finite precision, which might affect the accuracy for very large numbers or very high indices, though for most practical purposes, the precision is very high. You might also explore our Exponent Calculator for related calculations.

Frequently Asked Questions (FAQ)

What is the 1st root of a number?

The 1st root of a number ‘a’ is ‘a’ itself (a^1/1 = a).

Can the index ‘n’ be a fraction or decimal?

Yes, the index ‘n’ can be any positive real number. For example, a^1/2.5 is a valid operation, equivalent to a^2/5 or the 5th root of a².

What if the radicand is negative?

If the index ‘n’ is odd (3, 5, 7, etc.), a negative radicand will have a negative real root. If ‘n’ is even (2, 4, 6, etc.), a negative radicand does not have a real number root (it has complex roots).

How do I find the nth root without a calculator?

For simple cases (like the 4th root of 81), you might guess and check or use factorization. For more complex numbers, you would typically use logarithms or numerical methods like the Newton-Raphson method, which are what calculators employ.

Is the nth root the same as dividing by n?

No, finding the nth root is very different from dividing by n. The nth root is about finding a base that, when raised to the power n, gives the original number. Division is simply splitting a number into ‘n’ equal parts.

What is the difference between an Nth Root Calculator and a power calculator?

An Nth Root Calculator finds ‘x’ in xⁿ = a, while a power calculator finds ‘a’ in xⁿ = a given x and n. They are inverse operations related to exponents. See our Power and Root guide.

Can I find the root of 0?

Yes, the nth root of 0 is 0 for any positive index ‘n’.

What happens when n is very large?

As ‘n’ gets very large, the nth root of any positive number ‘a’ approaches 1 (if a > 0).

Related Tools and Internal Resources

Explore other calculators and resources that might be helpful: