About the Simplify Radicals Calculator
This simplify radicals calculator rewrites a root in simplest radical form. Enter the number under the radical (the radicand), choose square root, cube root or a higher index, and optionally a coefficient in front. The tool pulls every perfect-power factor out of the radical — so √72 becomes 6√2 and ∛54 becomes 3∛2 — and gives the decimal value too.
It is aimed at algebra and geometry students, and anyone checking answers involving the Pythagorean theorem, quadratic formula or distance formula where exact radical answers are expected. The factor table shows which prime factors were grouped and moved outside the root.
Negative radicands work for odd roots (∛−8 = −2). For a negative square root the result is written with the imaginary unit i, e.g. √−12 = 2i√3. Radicands up to 100,000,000 are supported.
How to use the simplify radicals calculator
- 1Enter the coefficient in front of the root, or leave it as 1.
- 2Enter the radicand — the whole number under the radical sign.
- 3Choose the type of root: square, cube or a higher index.
- 4Read the simplest radical form and its decimal value.
- 5Use the factor table to see which primes were pulled out.
Formula and method
A radical is in simplest form when the radicand has no factor that is a perfect n-th power (other than 1), there is no fraction under the root, and no root in a denominator. To get there, the radicand is broken into prime factors and every complete group of n identical primes is moved outside the radical as a single factor.
For √72: 72 = 2³ × 3². One pair of 2s and one pair of 3s come out as 2 × 3 = 6, leaving a single 2 inside, so √72 = 6√2. Any coefficient already in front is multiplied by the number that comes out.
- n
- Index of the root (2 = square root, 3 = cube root)
- a
- Factor taken out of the radical
- b
- Factor left inside (no perfect n-th power factors)
Worked examples
Simplify √72
72 = 36 × 2 and 36 is a perfect square, so √72 = √36 × √2 = 6√2 ≈ 8.4853.
Simplify the cube root of 54
54 = 27 × 2 and 27 = 3³ is a perfect cube, so ∛54 = 3∛2 ≈ 3.7798.
Simplify 3√50
50 = 25 × 2, so √50 = 5√2. Multiplying by the coefficient 3 gives 15√2 ≈ 21.2132.
Square root of a negative number, √−12
√−12 = √−1 × √4 × √3 = i × 2 × √3, written 2i√3. The decimal value 3.4641 is the coefficient of i.
Frequently asked questions
What is simplest radical form?+
A radical is in simplest form when the number under the root has no perfect-square factor (for square roots) or perfect-cube factor (for cube roots) other than 1, there are no fractions under the root, and no radicals in the denominator. For example 6√2 is simplest form, but √72 is not.
How do you simplify a square root?+
Find the largest perfect square that divides the number, split the root into √(perfect square) × √(rest), then take the square root of the perfect square. √48 = √16 × √3 = 4√3.
How do you simplify a cube root?+
Look for perfect-cube factors such as 8, 27, 64 or 125. ∛40 = ∛8 × ∛5 = 2∛5. Any prime that appears three times in the factorization comes out of the cube root once.
Can you take the square root of a negative number?+
Not as a real number, but in the complex numbers √−1 = i. So √−25 = 5i and √−12 = 2i√3. Odd roots of negatives are real: ∛−27 = −3.
Is √2 already simplified?+
Yes. 2 is prime, so it has no perfect-square factor other than 1 and √2 cannot be simplified further. The same is true for the square root of any prime or any product of distinct primes, like √30.