Radical Calculator

A radical like √72 usually should be simplified: pull out the largest perfect-square factor and 72 becomes 6√2. This calculator simplifies square roots to exact radical form with a decimal approximation, and can also evaluate any nth root directly.

Results

Formula

Use the product rule to extract perfect squares:
√(a × b) = √a × √b  →  √72 = √(36 × 2) = 6√2
If no factor larger than 1 is a perfect square, the radical is already simplified.

How to use this calculator

  1. Choose “Simplify a square root” or “Evaluate an nth root.”
  2. Enter the number under the radical (and the index for nth roots).
  3. Press Calculate for the exact form and the decimal value.
Radical Calculator — how the calculation works, illustratedΣ
Radical Calculator — how the calculation works, illustrated

Worked example

Simplify √72.

  1. Find the largest perfect-square factor of 72: 36.
  2. Rewrite: √72 = √(36 × 2) = √36 × √2.
  3. √36 = 6, so the result is 6√2 ≈ 8.485.

Result: √72 = 6√2 ≈ 8.485281.

Frequently asked questions

Why simplify radicals at all?

Simplest radical form is the exact, standard way to write irrational square roots — it makes answers comparable and further algebra much easier than a long decimal.

Can I take the square root of a negative number?

Not within the real numbers. Negative radicands need imaginary numbers (√−4 = 2i). Odd roots, like cube roots, do work for negatives: ∛−8 = −2.

Is √(a + b) equal to √a + √b?

No — that is a common mistake. Roots distribute over multiplication, never over addition.

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