Nth Root of a Number: How to Calculate

This calculator finds the nth root of a number — that is, the value that, when multiplied by itself n times, gives the original number. You enter the degree (n) and the radicand (the number under the root), and you get the exact root value as a decimal. It handles positive and negative bases, fractional roots, and shows the result rounded to your chosen precision.

Formula

The nth root of a number x is written as �√x and is defined by the equation:

y = x^(1/n)

where:

• x is the radicand (the number you are taking the root of),
• n is the degree (the root index), a positive integer,
• y is the result — the number such that yⁿ = x.

For example, the cube root of 27 is 3 because 3³ = 27. The fourth root of 16 is 2 because 2� = 16. In general, the nth root is the inverse operation of raising to the nth power.

Worked Example

Using the calculator defaults: n = 3 and x = 27.

1. Write the expression: ³√27 = 27^(1/3).
2. Find the number that, when cubed, equals 27: 3 × 3 × 3 = 27.
3. Therefore, ³√27 = 3.

The calculator returns 3.000000, matching the manual result.

How the nth Root Works

Raising a number to a fractional exponent is equivalent to taking a root. The key rule is:

x^(1/n) = �√x

For integer n, the nth root of x is the number y such that y multiplied by itself n times equals x. For example:

• Square root (n=2): √9 = 3 because 3² = 9.
• Cube root (n=3): ∛64 = 4 because 4³ = 64.
• Fourth root (n=4): â�´âˆš81 = 3 because 3â�´ = 81.

When n is even, the root of a positive number has two real solutions (one positive, one negative), but the principal root is the positive one. When n is odd, the root of a negative number is negative. For example, ∛(-8) = -2 because (-2)³ = -8. But √(-4) is not a real number because no real number squared equals -4.

How to Choose the Degree (n)

The degree n is the number of times the root is multiplied by itself. You choose it based on the problem:

• n = 2: square root — used in geometry (areas), physics, and statistics.
• n = 3: cube root — used in volume calculations, cubic equations, and some physics formulas.
• n = 4: fourth root — appears in quartic equations and some engineering contexts.
• Higher n: appear in higher-degree equations and advanced mathematics.

If the problem says “cube root�, n = 3. If it says “fifth root�, n = 5. The degree is always the small number in the root symbol.

Common Mistakes and Misconceptions

• Confusing root with division: The nth root is not x divided by n. For example, ³√27 is 3, not 9.
• Ignoring negative bases: If n is even and x is negative, the result is not a real number. The calculator will show an error or “NaNâ€� in that case.
• Mixing up even and odd roots: Even roots of positive numbers have two real solutions, but the principal (positive) root is used by default. Odd roots of negative numbers are negative.
• Rounding too early: In multi-step calculations, round only at the final step to avoid accumulating errors.

Real-World Uses of the nth Root

The nth root appears in many practical fields:

• Finance: Compound interest formulas use nth roots to find average growth rates over multiple periods.
• Engineering: Stress calculations and material properties sometimes involve cube roots or fourth roots.
• Physics: Wave equations and energy formulas may use square roots or cube roots.
• Computer graphics: Color correction and scaling often use roots to adjust brightness or size.
• Statistics: The geometric mean uses the nth root of a product of n numbers.

Limitations of the Calculator

This calculator works with real numbers. It does not compute complex roots (for even n and negative x). It also does not handle symbolic algebra — it gives a decimal approximation, not an exact radical form. For exact forms, use a computer algebra system. The calculator assumes n is a positive integer; fractional or negative n are not standard roots and may give unexpected results.

Frequently Asked Questions

What is the nth root of a number?

The nth root of a number x is the value y such that yⁿ = x. For example, the 5th root of 32 is 2 because 2â�µ = 32.

How do I find the nth root without a calculator?

For perfect powers, you can guess and check. For others, use logarithms: log(y) = (1/n) × log(x), then take the antilog. Or use the method of successive approximations (Newton’s method).

What is the difference between square root and nth root?

A square root is the nth root with n=2. The nth root generalizes this to any degree. For example, the square root of 16 is 4, but the 4th root of 16 is 2.

Can the nth root of a negative number be real?

Yes, if n is odd. For example, ∛(-27) = -3. If n is even, the result is not a real number (it is a complex number).

What does an nth root of 0 give?

The nth root of 0 is always 0, because 0ⁿ = 0 for any positive n.

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in n | Degree (n) | - | 3 | 0
in x | Radicand (x) | - | 27 | 0
out root | nth root | - | x^(1/n) | 6