Find The Nth Root Of A Number Calculator

Calculate the Nth Root of a Number (e.g., square root, cube root, etc.). Enter the number and the root index below.

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Roots Table

Root (n)	Value (n-th root of 27)
2	5.196152422706632
3	3
4	2.279507056954776
5	1.9331820449317622
10	1.3903891754189396

Table showing different roots of the number 27.

Root Value vs. Root Index (n) Chart

What is the Nth Root of a Number?

The Nth Root of a Number is a value that, when multiplied by itself ‘n’ times, equals the original number. For example, the 3rd root (or cube root) of 27 is 3 because 3 × 3 × 3 = 27. The most common root is the square root (where n=2), but we can find any nth root (3rd, 4th, 5th, etc.). The Nth Root of a Number is the inverse operation of raising a number to the power of ‘n’.

This concept is widely used in various fields like mathematics, engineering, finance (for compound interest over fractional periods), and physics. Anyone needing to reverse an exponentiation operation will use the Nth Root of a Number. For instance, if you know the volume of a cube and want to find its side length, you’d calculate the cube root of the volume.

Common misconceptions include thinking that only square and cube roots exist or are useful, or that the base number must always be positive. While taking an even root of a negative number results in a complex number (not a real number), odd roots of negative numbers are real and negative. Our Nth Root of a Number calculator focuses on real number results.

Nth Root of a Number Formula and Mathematical Explanation

The Nth Root of a Number ‘x’ for a root index ‘n’ is mathematically represented as:

ⁿ√x = x^1/n

Where:

• x is the base number (radicand).
• n is the root index (the degree of the root).
• ⁿ√ is the radical symbol, with ‘n’ indicating the root.
• x^1/n is the exponential form, where the root is expressed as a fractional exponent (1/n).

To calculate the Nth Root of a Number, you raise the number ‘x’ to the power of (1/n).

Variable	Meaning	Unit	Typical Range
x	The base number (radicand)	Unitless or units of the problem	Any real number (though even roots of negative numbers are not real)
n	The root index or degree	Unitless	n > 0, often integers but can be any positive real number
1/n	The fractional exponent	Unitless	Depends on n
^n√x or x^1/n	The nth root of x	Same units as x^1/n	Real or complex numbers

Variables used in the Nth Root of a Number calculation.

Practical Examples (Real-World Use Cases)

Example 1: Finding the Side of a Cube

Suppose you have a cube with a volume of 64 cubic meters. The volume (V) of a cube is given by side³. To find the length of a side, you need to calculate the cube root (3rd root) of the volume.

• Number (x): 64
• Root (n): 3

Using the Nth Root of a Number formula: side = 64^1/3 = 4 meters. So, the side of the cube is 4 meters.

Example 2: Geometric Mean

If an investment grew by 5% one year, 10% the next, and 8% the third year, the growth factors are 1.05, 1.10, and 1.08. To find the average annual growth factor (geometric mean), you multiply these factors and take the 3rd root of the product: (1.05 * 1.10 * 1.08)^1/3 = (1.2474)^1/3 ≈ 1.0765. This means an average annual growth of about 7.65%.

• Number (x): 1.2474
• Root (n): 3

The Nth Root of a Number here gives the geometric mean factor.

Check out our Geometric Mean Calculator for more.

How to Use This Nth Root of a Number Calculator

Using our Nth Root of a Number calculator is straightforward:

1. Enter the Number (x): Input the number for which you want to find the root into the “Number (x)” field. This is the base or radicand.
2. Enter the Root (n): Input the root index (e.g., 2 for square root, 3 for cube root) into the “Root (n)” field. This must be a positive number.
3. View Results: The calculator automatically displays the Nth Root of a Number (primary result), along with the base, root index, and the fractional exponent (1/n).
4. See Table and Chart: The table shows roots for different indices ‘n’ for your base ‘x’, and the chart visualizes how the root value changes.
5. Reset: Click “Reset” to return to the default values.
6. Copy Results: Click “Copy Results” to copy the main result and details to your clipboard.

The results help you understand how different roots affect the original number. For a base greater than 1, as ‘n’ increases, the nth root decreases towards 1. For a base between 0 and 1, as ‘n’ increases, the nth root increases towards 1.

Key Factors That Affect Nth Root of a Number Results

1. Base Number (x): The magnitude of the base number directly influences the magnitude of its root. Larger positive numbers will have larger roots (for a fixed n > 1).
2. Root Index (n): As the root index ‘n’ increases (for x > 1), the nth root value decreases. Conversely, as ‘n’ decreases towards 1, the root value approaches x.
3. Sign of the Base Number (x): If ‘x’ is positive, the nth root is positive. If ‘x’ is negative, only odd roots will yield a real, negative number. Even roots of negative numbers are not real numbers. Our Nth Root of a Number calculator highlights this by returning ‘NaN’ for even roots of negative numbers.
4. Whether n is an Integer or Fraction: While we often think of ‘n’ as an integer, the Nth Root of a Number can be calculated for fractional ‘n’ as well, corresponding to more complex exponents.
5. Magnitude of n compared to x: The relative sizes of x and n determine how quickly the root value changes as n changes.
6. Precision of Calculation: The number of decimal places used in the calculation can affect the precision of the resulting Nth Root of a Number, especially for irrational roots. Our tool uses standard JavaScript precision.

You might find our Exponent Calculator useful for related calculations.

Frequently Asked Questions (FAQ)

1. What is the 2nd root of a number?

The 2nd root is the square root. For example, the 2nd root of 9 is 3 because 3 * 3 = 9. Use our Square Root Calculator for this specific case.

2. What is the 3rd root of a number?

The 3rd root is the cube root. For example, the 3rd root of 8 is 2 because 2 * 2 * 2 = 8. See our Cube Root Calculator.

3. Can the root index ‘n’ be a fraction in the Nth Root of a Number?

Yes, ‘n’ can be a fraction. For example, if n = 2/3, you are calculating x^1/(2/3) = x^3/2.

4. What happens if the base number ‘x’ is negative?

If ‘x’ is negative and ‘n’ is an odd integer (1, 3, 5, …), the nth root is real and negative. If ‘x’ is negative and ‘n’ is an even integer (2, 4, 6, …), the real nth root does not exist (it’s a complex number). Our Nth Root of a Number calculator shows NaN for the latter.

5. What is the 1st root of a number?

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

6. Can I calculate the Nth Root of a Number equal to zero?

Yes, the nth root of 0 is 0 for any n > 0.

7. What if the root index ‘n’ is negative?

A negative root index implies 1 / (x^1/|n|). For example, x^1/-2 = 1 / x^1/2. Our calculator assumes n > 0 as per the standard definition of ⁿ√x.

8. How is the Nth Root of a Number related to logarithms?

They are related through exponents. If y = x^1/n, then log(y) = (1/n)log(x). You might be interested in our Logarithm Calculator.

Related Tools and Internal Resources

• Square Root Calculator: Specifically calculates the 2nd root.
• Cube Root Calculator: Specifically calculates the 3rd root.
• Exponent Calculator: Calculates x raised to the power of y.
• Logarithm Calculator: Calculates logarithms, the inverse of exponentiation.
• Power Calculator: Similar to the exponent calculator, finds the result of a base raised to an exponent.
• Radical Calculator: Simplifies radicals and finds roots.