Root Calculator (Radical) – Nth Root Solver

Compute square roots, cube roots, and general nth roots of real numbers. Check when a real root exists, see the result as a decimal and, when possible, as a simplified radical.

  • Handles any real radicand and integer index n (2, 3, 4, …).
  • Warns when no real root exists (e.g., even root of a negative number).
  • Shows radical notation and simplified radical form for suitable integers.

Radical (Root) Calculator

You can enter positive or negative real numbers.

Use n = 2 for square root, n = 3 for cube root, etc. n must be a non-zero integer.

Choose the number of decimal places (0–15) for the numeric result.

Numeric result

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Radical notation

Expression: –

LaTeX form: \(\sqrt[n]{x}\)

Simplified radical (for suitable integers)

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For integer radicands with moderate size, we factor the number and pull out perfect nth powers.

What is a root (radical)?

Given a real number x and a positive integer n, the nth root of x is a real number y such that \( y^n = x \). We write:

\[ y = \sqrt[n]{x} \quad \Longleftrightarrow \quad y^n = x \]

Special cases appear all the time in algebra and calculus:

  • \(\sqrt{x}\) is the square root of \(x\) (here \(n = 2\)).
  • \(\sqrt[3]{x}\) is the cube root of \(x\) (here \(n = 3\)).
  • More generally, \(\sqrt[n]{x}\) denotes the nth root of \(x\).

Roots and fractional exponents

Roots can be written using fractional exponents. For any real \(x \ge 0\) and integer \(n > 0\):

\[ \sqrt[n]{x} = x^{1/n} \]

This identity is extremely useful when simplifying expressions, taking derivatives, or working with numerical methods. For example:

  • \(\sqrt{x} = x^{1/2}\)
  • \(\sqrt[3]{x^2} = x^{2/3}\)
  • \(\sqrt[4]{x^3} = x^{3/4}\)

Domain of the nth root function (real numbers)

Over the real numbers, not every combination of index \(n\) and radicand \(x\) gives a valid root:

  • If \(n\) is even (2, 4, 6, …), you must have \(x \ge 0\) to get a real root. For example, \(\sqrt{-4}\) has no real value.
  • If \(n\) is odd (3, 5, 7, …), any real \(x\) works. For instance, \(\sqrt[3]{-8} = -2\) is perfectly valid.
  • The index \(n\) cannot be zero. The expression \(\sqrt[0]{x}\) is undefined.

The calculator checks these conditions for you and will warn when the requested real root does not exist.

Simplifying radicals: pulling out perfect powers

When the radicand is an integer and not too large, you can often write the root in a simplified radical form by factoring out perfect nth powers.

For example, consider the square root of 72:

\[ 72 = 36 \cdot 2 = 6^2 \cdot 2 \quad \Rightarrow \quad \sqrt{72} = \sqrt{6^2 \cdot 2} = 6\sqrt{2}. \]

Our calculator automates this process for integer radicands with moderate size (e.g., \(|x| \le 10^9\)) and integer indices \(2 \le n \le 10\), showing a cleaned-up expression like \(3\sqrt[3]{2}\) or \(5\sqrt{7}\) when possible.

Worked examples

Example 1 – Square root of 72

Input \(x = 72\), \(n = 2\). The calculator:

  • Returns the decimal value \(\sqrt{72} \approx 8.485281\ldots\).
  • Shows the expression \(\sqrt{72}\).
  • Simplifies this to \(6\sqrt{2}\).

Example 2 – Cube root of −125

Input \(x = -125\), \(n = 3\). Since the index is odd, a real root exists:

\[ \sqrt[3]{-125} = -5, \quad \text{because } (-5)^3 = -125. \]

The calculator returns −5 as the exact numeric result and indicates that this is a valid real root.

Example 3 – Fourth root of −16

Input \(x = -16\), \(n = 4\). Because \(n\) is even and the radicand is negative, there is no real fourth root. The calculator will display a domain warning and will not produce a real number.

FAQ – Root & radical calculator

What is the difference between a square root and an nth root?

A square root is a specific case of an nth root with \(n = 2\). The nth root generalizes this idea: instead of asking “what number squared equals x?”, you ask “what number raised to the power n equals x?”.

Why do I get an error for negative numbers with even index?

Over the real numbers, even roots of negative numbers are not defined. For example, there is no real number whose square is −9. To work with these cases, you would need complex numbers (e.g., \(\sqrt{-9} = 3i\)), which this calculator does not handle.

Why does the simplified radical sometimes stay the same as the original?

Not every integer has a factor that is a perfect nth power. If no perfect nth powers can be factored out, or if the number is too large for safe factorization, the simplest radical form is just the original expression.

What level of precision is appropriate for scientific work?

It depends on your application and the precision of your input data. The calculator lets you choose up to 15 decimal places, but in practice you should match the number of significant figures to the reliability of your measurements or constants.