About the Simplest Radical Form Calculator
Teachers and textbooks usually ask for answers in simplest radical form, not as decimals. √72 is correct but not simplified; 6√2 is the same number written so that nothing more can be taken out of the root. Simplest form is exact, easier to compare and easier to combine with other radicals, which is why it is the standard way to present answers in algebra, geometry and trigonometry.
This simplest radical form calculator simplifies a square root, a cube root or any root up to the tenth. It shows the prime factorisation, which factors come out of the root and which stay in, handles a coefficient in front, rationalises roots of fractions so no root remains in a denominator, and gives the decimal value as a check.
How to Use the Simplest Radical Form Calculator
Enter the coefficient in front of the root, or 1 if there is none.
Enter the number under the root, the radicand. It can be negative for odd roots, or for square roots, which give an answer with i.
Enter a denominator under the root if the radicand is a fraction, as in √(5/8). Otherwise leave it at 1.
Enter the index: 2 for a square root, 3 for a cube root, and so on.
The Formulas
write the radicand as a product of primes: r = p1^e1 × p2^e2 × …
ⁿ√r = (p1^⌊e1/n⌋ × p2^⌊e2/n⌋ × …) × ⁿ√(p1^(e1 mod n) × p2^(e2 mod n) × …)
for a fraction: ⁿ√(a/b) = ⁿ√(a × bⁿ⁻¹) / bStep-by-Step Example
Simplify √72.
72 = 2³ × 3²
pairs: one pair of 2s and one pair of 3s come out → 2 × 3 = 6
left inside: one 2
√72 = 6√2 ≈ 8.48528Simplify ∛54. 54 = 2 × 3³. The group of three 3s comes out as 3 and the 2 stays inside, so ∛54 = 3∛2 ≈ 3.77976.
Simplify √(5/8). Rationalise first: √(5/8) = √(5 × 8)/8 = √40/8. Then 40 = 2³ × 5, so √40 = 2√10, and 2√10/8 = √10/4 ≈ 0.790569.
Why Prime Factors Work
A root undoes a power. The square root of p² is p, and the cube root of p³ is p. Breaking the radicand into primes shows exactly how many complete powers of the index it contains. Each complete group comes out as a single factor; whatever does not make a complete group stays inside. Because every whole number has only one prime factorisation, the result is unique: there is exactly one simplest form.
You can also simplify by spotting the largest perfect square factor directly. 72 = 36 × 2, and √36 = 6, giving 6√2 in one step. That is quicker when you recognise the factor, but the prime method never misses one. A common slip is to take out a smaller square, such as 72 = 4 × 18 giving 2√18, and stop there, even though 18 still contains the square 9.
Coefficients and Fractions
A coefficient in front multiplies whatever comes out of the root. 3√50 = 3 × 5√2 = 15√2. When there is a denominator, the factor that comes out may cancel with it, as 2√10/8 reduced to √10/4 above. The calculator reduces the fraction in front to lowest terms.
Rationalising the denominator is part of simplest form because a root in the denominator makes expressions harder to compare and add. For square roots, multiply the top and bottom by the root in the denominator: 1/√2 = √2/2. For higher roots you multiply by enough copies to make a perfect power, which is why ⁿ√(a/b) becomes ⁿ√(a·bⁿ⁻¹)/b.
Negative Radicands
Odd roots of negative numbers are real and negative, because a negative number cubed is negative: ∛−54 = −3∛2. Square roots of negative numbers are not real; they are imaginary, written with i, where i² = −1. So √−12 = i√12 = 2i√3. Fourth roots, sixth roots and other even roots of negative numbers are also not real, and the calculator refuses them rather than guessing a complex form.
Where Simplest Radical Form Appears
Radicals come up throughout geometry and trigonometry. The diagonal of a square with side 1 is √2, the height of an equilateral triangle with side 2 is √3, and the exact values sin 45° = √2/2 and sin 60° = √3/2 are all in simplest form. The quadratic formula often produces answers like (−2 ± √12)/2, which simplify to −1 ± √3. Distance and Pythagorean theorem problems give results such as √45 = 3√5.
Estimating Roots to Check Your Work
A quick estimate catches mistakes. √72 lies between √64 = 8 and √81 = 9, a little above 8.5, and 6√2 ≈ 6 × 1.414 = 8.485 fits. If you had written 3√8, the estimate 3 × 2.83 = 8.49 also fits, which shows that a decimal check alone cannot tell you whether the form is fully simplified; 8 still contains the square 4.
Understanding Your Result
The headline shows the original root and its simplest radical form.
The decimal line gives the value to about eight significant figures.
The check line confirms that no perfect power remains under the root, or that the radicand was a perfect power.
The steps show any rationalising, the prime factorisation and the groups taken out of the root.
When Should You Use This Calculator?
Use it to simplify square and cube roots for homework and exams.
Use it to rationalise roots of fractions.
Use it to check exact answers from the quadratic formula, the Pythagorean theorem or trigonometry.
Common Mistakes
Taking out a square that is too small. √72 = 2√18 is not finished; 18 still contains 9.
Leaving a root in the denominator. Rationalise: 1/√2 = √2/2.
Splitting a sum. √(9 + 16) is 5, not √9 + √16 = 7.
Treating even roots of negatives as real. √−4 = 2i, not −2.