Radical Calculator

Enter a radicand and root index to simplify a radical (ⁿ√b) into simplest form, see its exact decimal value, and view the equivalent exponent notation.

Quick Facts

Radical to exponent
ⁿ√b = b^(1/n)
A radical is just a fractional exponent written a different way.
Product rule
ⁿ√(a·b) = ⁿ√a · ⁿ√b
Splitting factors this way is how perfect powers get pulled out.
Perfect squares
1, 4, 9, 16, 25, 36, 49, 64...
Recognizing these speeds up simplifying square roots by hand.
Even-index rule
Even n requires b ≥ 0
Even roots of negative numbers are not real numbers.

Your Results

Calculated
Simplified Radical Form
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coefficient × ⁿ√(reduced radicand)
Decimal Value
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Numeric approximation
Exponent Form
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Radical written as a fractional power
Simplification Note
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What was factored out, if anything

Ready

Enter a radicand and root index, then press Calculate.

Formula and Method for the Radical Calculator

A radical expression ⁿ√b (read "the nth root of b") asks: what number, raised to the power n, equals b? The small number n written to the upper left of the root symbol is the index; when no index is written, it is assumed to be 2 (a square root). This calculator simplifies ⁿ√b into simplest radical form by factoring out every perfect nth-power factor, and also reports the exact decimal value and the equivalent exponent notation b^(1/n).

How the simplification works

The calculator finds the prime factorization of the radicand. For every prime factor p raised to an exponent e, it pulls out ⌊e ÷ n⌋ complete groups of n copies of p — each group becomes one factor of p outside the root — and leaves the remainder (e mod n) copies of p inside the root. For example, 72 = 2³ × 3²; with index n = 2, the factor 2³ contributes one pair (leaving one 2 inside) and 3² contributes one full pair (leaving nothing inside), so √72 = √(2³ × 3²) = 6√2. An optional coefficient simply multiplies the factor that ends up outside the root. This method requires a whole-number radicand and a whole-number index of 2 or greater; negative radicands are only valid — and real — when the index is odd.

Common mistakes

  • Adding radicals like ordinary numbers: a√c + b√c = (a + b)√c only when the radicand c is identical; √2 + √3 cannot be combined into √5.
  • Forgetting the even-root restriction: ⁿ√b is not a real number when n is even and b is negative (for example, √-4 has no real square root).
  • Stopping the simplification early: a radical is only in "simplest form" once the radicand has no remaining factor that is itself a perfect nth power (other than 1).

Real-world applications

  • Geometry: the Pythagorean theorem and the distance formula routinely produce answers in radical form (a diagonal of length √72 simplifies to 6√2).
  • Physics and engineering: root-mean-square values, resonance frequencies, and free-fall time all involve roots that are cleaner to communicate in simplified radical form.
  • Algebra and standardized tests: simplifying radicals is required for solving quadratic equations, since the quadratic formula contains a √(b² - 4ac) term.

Frequently Asked Questions

What does it mean to simplify a radical?
Simplifying a radical means rewriting ⁿ√b so the radicand no longer contains a factor that is a perfect nth power. For example, √72 simplifies to 6√2 because 72 = 36 × 2 and 36 is a perfect square (6²).
Can the radicand be negative?
Yes, but only when the index is odd. An odd root of a negative number is a real, negative number — for example, ³√-8 = -2 — while an even root of a negative number, like √-4, is not a real number.
How do I simplify a cube root or higher root?
Use the same idea as a square root: factor the radicand into primes and pull out every group of n identical prime factors for an index-n root. For a cube root, three identical prime factors combine into one factor outside the root; for example, ³√54 = ³√(27 × 2) = 3·³√2.
What is the difference between radical form and exponent form?
They are two notations for the same value: ⁿ√b is identical to b^(1/n). Exponent form is often easier to manipulate algebraically, while radical form is the conventional way to write exact answers in geometry and algebra.

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