Radical Calculator - Any nth Root Solver

A radical calculator is a math tool that solves the relationship r^n = x, where r is the principal nth root, n is the index that names the degree of the root, and x is the radicand under the radical sign. Use it to find any nth root, recover the original radicand from a known root and index, or estimate the index that connects the two.

• • Square, cube, and higher roots: Find the square root, cube root, fourth root, or any positive integer root of a positive number in one step.
• • Reverse the radical: Recover the radicand from a known result and index by raising the result to the power n, useful when checking factoring or geometry work.
• • Find the missing index: Estimate the degree of the radical when the radicand and the result are known, which helps when simplifying nested radicals or logs.
• • Compare radical and exponent form: Display the same value as both a radical and a fractional exponent so the two notations can be checked against each other.

The radical sign was introduced to avoid writing fractional exponents by hand, and the small number above the radical symbol names the degree of the root. When the index is 2 the result is the familiar square root; when it is 3 the result is the cube root; and for any other positive n the result is the general nth root. The radical calculator implements all three of those cases plus the reverse directions so the same page can answer most root questions a student or working professional needs.

Pair the calculator with the exponential notation calculator when you also need to express very large or very small numbers in compact e-notation alongside the radical form.

The calculator reads the chosen mode plus two of the three values (radicand, index, and result) and computes the third using a single exponent or logarithm operation.

• Radicand (x): The non-negative number under the radical. It is the value being rooted.
• Index (n): The positive root degree written above the radical. n=2 is a square root and n=3 is a cube root.
• Result (r): The principal nth root, the non-negative number r such that r raised to the index equals the radicand.

The relationship r^n = x is invertible on the positive real line, which is why the same calculator can answer the question in either direction. The find-the-radicand direction uses the familiar power rule r^n, while the find-the-index direction uses the change-of-base formula n = log(x) / log(r). That second direction is the same algebra used to convert between logarithm bases, so the underlying identity is well established in any pre-calculus reference.

According to Wolfram MathWorld, the radical of a number is the root defined by the equation r^n = x, with the principal nth root written as x^(1/n).

According to Khan Academy, the principal nth root of a non-negative number x is the non-negative number r such that r^n equals x, and a radical can always be rewritten as a fractional exponent with denominator equal to the index.

When you need to see the same number as a fractional exponent, the fractional exponent calculator can show the x^(1/n) form with the same precision you choose here.

Four short ideas explain why a radical behaves the way it does, and each one is also relevant to the rest of pre-algebra and algebra work.

Two radicals that look different can represent the same number when the index and the radicand match, which is why simplifying a radical means rewriting the radicand so the index and the number under the sign are in their simplest form. A useful trick is to extract perfect nth powers, the way 192^(1/3) becomes 4 * 3^(1/3) because 192 = 64 * 3 and 64 is a perfect cube.

When a problem mixes radicals with absolute values, the absolute value calculator helps confirm whether a sign has flipped during a principal-root step.

The calculator accepts any two of the three values, then solves for the third based on the chosen mode.

1. 1 Choose what to solve for: Use the Solve For select to pick the nth root, the radicand, or the index as the computed output.
2. 2 Enter the radicand and index: Type the number under the radical sign and the small index that names the root. Leave the third field empty or let the calculator overwrite it.
3. 3 Adjust the precision: Set the number of decimal places shown for the computed value. Use a higher precision for engineering or scientific reporting and a lower precision for quick mental checks.
4. 4 Read the result panel: The Computed Value, Fractional Exponent, Cross Check, and Perfect Root flag update on every change so the answer is always available.
5. 5 Reverse the calculation: Switch the mode to Find the radicand or Find the index to recover the missing value from the same three inputs.

Suppose you need the side length of a cube with a volume of 729 cubic units. Set the mode to Find the nth root, enter 729 as the radicand and 3 as the index, and the result panel returns 9, with the cross check 9^3 = 729 confirming the answer.

The main benefit is having a single page that answers radical questions in any direction and shows the same value in two equivalent notations.

• • Three directions in one tool: Find the nth root, recover the radicand, or estimate the index from the same set of inputs without rebuilding the formula each time.
• • Cross-checks built in: The cross check raises the result to the index so the original radicand can be matched against the recovered value.
• • Two equivalent notations: Radical form and fractional-exponent form are shown side by side, which is helpful when an answer key uses a different convention.
• • Perfect-root flag: The perfect-root flag points out clean integer results so you can spot perfect squares, cubes, and higher powers without factoring by hand.
• • Adjustable precision: Precision can be increased for engineering or scientific work and lowered for quick mental math, all from the same calculator.

If the answer needs to be reused inside a larger expression, the fractional-exponent output is the easier form to combine with other powers, while the radical output is the easier form to drop back into a textbook solution. The calculator shows both so the choice stays with the user.

When the problem is part of a quadratic or polynomial equation, the quadratic formula calculator can take the same radical result and place it inside the full quadratic-formula solution.

The reliability of the result depends on the inputs the user supplies and on a few well-known constraints of real-valued radicals.

• • The calculator returns the principal real root only, so even-index radicals of negative numbers are reported as undefined rather than a complex conjugate pair.
• • The find-the-index direction is an estimate based on the change-of-base formula, so the result rounds to the same decimal precision shown in the precision input and may need a small correction in symbolic work.
• • Radicals of mixed-sign sums, like sqrt(a + b), are not reduced to a single closed form by this tool because that simplification depends on the specific values of a and b.

These limits matter most when a problem is being checked against a symbolic answer in a textbook, and a quick scan of the cross-check row confirms whether the chosen mode and inputs are consistent with the original radicand.

According to Britannica, a radical is the symbol that indicates a root of a number, with a small index above the radical sign naming the degree of the root.

A general nth-root reference is a good companion when an odd-index root of a negative number is needed: the root calculator includes that case and lets you compare the principal-root result with the full nth-root result.