Root Calculator

Calculate the principal real nth root of a number, verify it by raising the result back to the selected power, and export the current calculation to a validated Excel workbook.

Domain Real Index 4th Status Exact integer root

Inputs

Real-number mode

Radicand (a)

A real number in decimal or scientific notation. Negative values require an odd root index.

Root index (n)

A whole number from 1 to 100,000. Use 2 for square root, 3 for cube root, and so on.

Model: b = a^(1/n), where b^n = a. For a negative radicand and odd n, the real root is negative.

Live results

Principal real root

Principal root

The 4th root of 81 is 3.

Reciprocal exponent

0.25

Power check

Absolute error

The 4th root of 81 is 3; raising 3 to the 4th power returns 81.

Calculation checkpoints

Built from the same canonical model

Quantity	Current value	Interpretation
Radicand (a)	81	The number under the radical.
Root index (n)	4	The power that reverses the root.
Reciprocal exponent (1/n)	0.25	The equivalent exponent applied to a.
Principal root (b)	3	The selected real value satisfying b^n = a.
Power check (b^n)	81	Should reproduce the radicand within floating-point precision.
Absolute error	0	The absolute difference between the power check and radicand.

The table reports the current model values. Very small nonzero error can appear for irrational roots because browser arithmetic uses finite binary floating-point precision.

How to use the root calculator

What this calculator does

This calculator finds the principal real nth root of a number. In mathematical terms, it solves for b in the identity bⁿ = a, where a is the radicand and n is the root index. It also shows the equivalent reciprocal exponent, raises the result back to the selected power, and reports the absolute numerical error. The tool works in the real-number system; it does not list the multiple complex roots that a nonzero number may have.

When to use it

Use the calculator to check square, cube, fourth, or higher roots; reverse a known power; estimate a side length from an area or volume; or verify a radical calculation made by hand. It is also useful when converting between radical notation and rational-exponent notation. OpenStax's explanation of the nth root and radical index provides a concise mathematical foundation for these operations.

How to calculate

The calculator opens with the demonstration a = 81 and n = 4, so meaningful results and a validated example XLSX workbook are available immediately.
Replace Radicand (a) with the number whose root you need. Decimal and scientific notation are accepted, such as 72, 0.008, or 1e12. Commas, percent signs, currency symbols, and mixed text are rejected rather than silently reinterpreted.
Replace Root index (n) with a whole number from 1 through 100,000. Enter 2 for a square root, 3 for a cube root, 4 for a fourth root, and so on.
Read the Principal root first, then use Power check and Absolute error to see how closely the rounded machine result reconstructs the original radicand.
Select Download Excel to export the current inputs and results. Select Reset to clear the demonstration values, results, table, and workbook state. After Reset, Excel export remains disabled until both required inputs form a complete valid real-root calculation again.

Input guide

Radicand (a) is required and accepts a finite real number in U.S.-style decimal notation, optionally with a scientific exponent. A realistic entry is 72. Increasing a positive radicand increases its positive principal root for a fixed index. Zero is valid and has root zero. A negative radicand is valid only when the index is odd; for example, the cube root of -8 is -2. A common mistake is entering a decimal comma such as 1,5, which is rejected to avoid interpreting it as 15.

Root index (n) is required and must be a positive integer from 1 to 100,000. A realistic entry is 3. For a positive radicand greater than 1, a larger index generally moves the root closer to 1. For values between 0 and 1, a larger index also moves the result toward 1 from below. Fractions, zero, negative values, and even indices paired with a negative radicand are invalid in this real-number calculator.

Output guide

Principal root is the principal real value b. For positive a it is nonnegative; for negative a with odd n it is negative. Reciprocal exponent is exactly 1/n and shows the equivalent power in a^1/n. Power check is bⁿ; it should agree with a. Absolute error is |bⁿ - a| and is normally zero or very small. The header's Domain pill confirms real-number mode, Index names the selected order, and Status distinguishes an exact integer root from a decimal approximation. The checkpoint table repeats the radicand, index, reciprocal exponent, principal root, power check, and error from the same model so they can be audited together.

Worked example

With the startup values a = 81 and n = 4, the calculator evaluates 81^1/4. Because 3 × 3 × 3 × 3 = 81, the Principal root is exactly 3. The reciprocal exponent is 0.25, the power check is 81, and the absolute error is 0. Those same typed values appear in the first-open workbook.

How nth roots work

An nth root reverses exponentiation. If bⁿ = a, then b is an nth root of a. The notation can be written as a radical or as a rational exponent:

ⁿ√a = a^(1/n)

The index determines the domain behavior over real numbers. An odd index accepts every real radicand because a negative base raised to an odd power remains negative. An even index cannot produce a negative value from a real base, so a negative radicand has no real even root. The broader relationship between radicals and fractional powers is covered in OpenStax's section on radicals and rational exponents.

Principal roots, exact roots, and approximations

Positive numbers can have two real square-root solutions to the equation x² = a, but the radical symbol conventionally returns the nonnegative principal square root. Higher even roots follow the same principal-value convention. Odd roots have one real value for every real radicand. Wolfram MathWorld's definition of the nth root summarizes the inverse relationship between powers and roots.

An exact integer root occurs when the radicand is a perfect nth power, such as 81 = 3⁴ or -125 = (-5)³. Most roots are irrational and must be approximated. For example, √2 does not terminate or repeat as a decimal. The displayed result therefore uses a compact precision strategy: ordinary-sized values are shown with up to twelve decimal places, while extremely large or small values use scientific notation.

Interpret the error correctly: a tiny nonzero absolute error usually reflects floating-point rounding, not a different mathematical answer. Compare the error with the size of the radicand rather than treating every nonzero digit as a failure.

Common mistakes

Using an even index for a negative radicand while expecting a real result.
Entering a fractional index when the intended operation is a standard nth root with integer order.
Confusing the root index n with the reciprocal exponent 1/n.
Rounding the root too early, then obtaining a noticeably different power check.
Assuming the principal root lists every algebraic or complex solution. It reports one conventional real value.