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Quick answer
A radical is in simplest radical form when no perfect square (for √) or perfect cube (for ∛) is left under the root, no fraction is under the root, no radical is in a denominator, and the index is as small as possible. To get there, factor the radicand and take out the largest perfect power: √72 = √(36 × 2) = 6√2, √(3/8) = √6/4, √(−2) = i√2.
Enter any radical — a whole number like 72, a fraction like 3/8, a negative like −2, or a monomial like 128u⁴ — with any index. You get the exact answer, its decimal value, the prime factorization, entire-radical and exponent forms, and every step. Other tabs add and multiply radicals, turn rational exponents into radical form, and recover a radical from a calculator decimal.
Last updated October 2, 2026. Checked against OpenStax Intermediate Algebra 2e and Paul’s Online Math Notes. · simplest radical form · add and subtract radicals
Whole number, fraction (3/8), negative (−2), decimal or monomial (128u^4)
The 4 in 4∛24; for 5/√13 use 5 and radicand 1/13
The 6 in √180 / 6
Simplest radical form
√72 = 6√2
Decimal value ≈ 8.4853
Prime factorization
72 = 2³ × 3²
Largest perfect square factor
36
Entire radical form
√72
Exponent form
6 · 2^(1/2)
LaTeX
6\sqrt{2}
Step-by-step
Radical form writes a root with the radical sign. In a·ⁿ√b, n is the index (2 for a square root, usually left off), b is the radicand, and a is the coefficient. The same number in exponent form is a·b^(1/n); in general b^(m/n) = ⁿ√(bᵐ), so 2^(3/4) = ∜(2³).
Simplest radical form (also called simplified or simple radical form) is the tidiest exact way to write that radical. It is what algebra and geometry teachers mean by "express in simplest radical form", and it is the form SAT, ACT and state-test answer choices use — for example the diagonal of a square with side 6 is 6√2, not √72 or 8.485.
| Rule | Not simplest | Simplest |
|---|---|---|
| No perfect nth power factor under the root | √72 | 6√2 |
| Variable exponents smaller than the index | √(y⁷) | y³√(y) |
| Exponents share no factor with the index | ⁹√(x⁶) | ∛(x²) |
| No fraction under the radical | √(3/8) | √6/4 |
| No radical in a denominator | 5/√13 | 5√13/13 |
The method rests on the product rule ⁿ√(ab) = ⁿ√a · ⁿ√b and the quotient rule ⁿ√(a/b) = ⁿ√a / ⁿ√b. Split off the biggest piece whose root is a whole number, and leave the rest under the radical.
1. Factor the radicand
Write the number under the root as a product of primes, e.g. 72 = 2³ × 3².
2. Find the largest perfect power
Group primes into sets of n (the index). For √72, 2² × 3² = 36 is the largest perfect square factor.
3. Take it outside the radical
Its root moves outside: √72 = √36 × √2 = 6√2. Multiply by any coefficient already in front.
4. Clear fractions and denominators
Multiply inside the radical (or top and bottom) so the denominator is a perfect power: √(3/8) = √6/4, 5/√13 = 5√13/13.
5. Reduce the index
If every exponent under the root shares a factor with the index, divide it out: ∜9 = √3.
6. Check
No perfect nth power, no fraction under the root, no radical in a denominator, smallest index — the radical is in simplest form.
Square root: √180
180 = 2² × 3² × 5
largest perfect square = 36 = 6²
√180 = √36 · √5 = 6√5 ≈ 13.4164
Cube root: 4∛24
24 = 2³ × 3
largest perfect cube = 8 = 2³
4∛24 = 4 · 2 · ∛3 = 8∛3 ≈ 11.538
Green simplifies, blue is a perfect square, white is already in simplest radical form.
Prime factorization, largest perfect square and the answer for the radicals people look up most.
| Radical | Prime factorization | Largest perfect square | Simplest radical form | Decimal |
|---|---|---|---|---|
| √72 | 72 = 2³ × 3² | 36 | 6√2 | 8.4853 |
| √24 | 24 = 2³ × 3 | 4 | 2√6 | 4.899 |
| √12 | 12 = 2² × 3 | 4 | 2√3 | 3.4641 |
| √8 | 8 = 2³ | 4 | 2√2 | 2.8284 |
| √45 | 45 = 3² × 5 | 9 | 3√5 | 6.7082 |
| √50 | 50 = 2 × 5² | 25 | 5√2 | 7.0711 |
| √54 | 54 = 2 × 3³ | 9 | 3√6 | 7.3485 |
| √99 | 99 = 3² × 11 | 9 | 3√11 | 9.9499 |
| √128 | 128 = 2⁷ | 64 | 8√2 | 11.3137 |
| √136 | 136 = 2³ × 17 | 4 | 2√34 | 11.6619 |
| √180 | 180 = 2² × 3² × 5 | 36 | 6√5 | 13.4164 |
| √242 | 242 = 2 × 11² | 121 | 11√2 | 15.5563 |
| √297 | 297 = 3³ × 11 | 9 | 3√33 | 17.2337 |
| √1020 | 1020 = 2² × 3 × 5 × 17 | 4 | 2√255 | 31.9374 |
| √14436 | 14436 = 2² × 3² × 401 | 36 | 6√401 | 120.1499 |
| √161 | 161 = 7 × 23 | 1 (none) | √161 | 12.6886 |
| √185 | 185 = 5 × 37 | 1 (none) | √185 | 13.6015 |
| √286 | 286 = 2 × 11 × 13 | 1 (none) | √286 | 16.9115 |
| √330 | 330 = 2 × 3 × 5 × 11 | 1 (none) | √330 | 18.1659 |
Take out perfect cubes (8, 27, 64, 125…) or 4th powers (16, 81, 256…).
| Radical | Factor | Simplest form | Decimal |
|---|---|---|---|
| ∛16 | 8 × 2 | 2∛2 | 2.5198 |
| ∛24 | 8 × 3 | 2∛3 | 2.8845 |
| ∛32 | 8 × 4 | 2∛4 | 3.1748 |
| ∛40 | 8 × 5 | 2∛5 | 3.42 |
| ∛54 | 27 × 2 | 3∛2 | 3.7798 |
| ∛72 | 8 × 9 | 2∛9 | 4.1602 |
| ∛81 | 27 × 3 | 3∛3 | 4.3267 |
| ∛108 | 27 × 4 | 3∛4 | 4.7622 |
| ∛128 | 64 × 2 | 4∛2 | 5.0397 |
| ∛135 | 27 × 5 | 3∛5 | 5.1299 |
| ∛192 | 64 × 3 | 4∛3 | 5.769 |
| ∛250 | 125 × 2 | 5∛2 | 6.2996 |
| ∛500 | 125 × 4 | 5∛4 | 7.937 |
| ∛(−54) | 27 × 2 | −3∛2 | −3.7798 |
| ∛(−297) | 27 × 11 | −3∛11 | −6.6719 |
| ∜32 | 16 × 2 | 2∜2 | 2.3784 |
| ∜48 | 16 × 3 | 2∜3 | 2.6321 |
| ∜80 | 16 × 5 | 2∜5 | 2.9907 |
| ∜162 | 81 × 2 | 3∜2 | 3.5676 |
No fraction under the root and no root in the denominator.
| Expression | Simplest form | Decimal |
|---|---|---|
| √(3/8) | √6/4 | 0.6124 |
| 5/√13 | 5√13/13 | 1.3868 |
| √(1/2) | √2/2 | 0.7071 |
| √(2/3) | √6/3 | 0.8165 |
| 3/√6 | √6/2 | 1.2247 |
| √180 / 6 | √5 | 2.2361 |
| −5√(11/49) | −5√11/7 | −2.369 |
| √(5/12) | √15/6 | 0.6455 |
| 2/∛4 | ∛2 | 1.2599 |
| √(21/10) | √210/10 | 1.4491 |
Even roots use i = √(−1); odd roots stay real and negative.
| Expression | Simplest form | Value |
|---|---|---|
| √(−2) | i√2 | 1.4142i |
| √(−3) | i√3 | 1.7321i |
| √(−10) | i√10 | 3.1623i |
| √(−72) | 6i√2 | 8.4853i |
| √(−297) | 3i√33 | 17.2337i |
| ∛(−54) | −3∛2 | −3.7798 |
| ∛(−297) | −3∛11 | −6.6719 |
Divide each exponent by the index: quotient outside, remainder inside (variables positive).
| Expression | Simplest radical form |
|---|---|
| ∛(128u⁴) | 4u∛(2u) |
| ∜(80x¹²z¹⁹) | 2x³z⁴∜(5z³) |
| ∛(64x¹²) | 4x⁴ |
| √(18x⁶y¹¹) | 3x³y⁵√(2y) |
| ∜(32x⁹y⁵z¹²) | 2x²yz³∜(2xy) |
| √(y⁷) | y³√(y) |
| ⁹√(x⁶) | ∛(x²) |
| √(x⁴y²) | x²y |
Simplify each term first; only like radicals combine under + and −.
| Expression | Simplest form | Decimal |
|---|---|---|
| 5√28 − 7√7 | 3√7 | 7.9373 |
| −6√80 − √5 | −25√5 | −55.9017 |
| 2√12 + 3√27 | 13√3 | 22.5167 |
| 10√72 + 7∛32 | 60√2 + 14∛4 | 107.0764 |
| √8 × √24 | 8√3 | 13.8564 |
| √5 × √15 | 5√3 | 8.6603 |
| 9√3 × √3 | 27 | 27 |
| 7√6 × 7√14 | 98√21 | 449.0924 |
| √(−2) × √(−3) | −√6 | −2.4495 |
| ∛(9x²) × ∛(6x²) | 3x∛(2x) | — |
| √2 × ∛2 | ⁶√32 | 1.7818 |
| 3√2 ÷ √6 | √3 | 1.7321 |
a^(m/n) = ⁿ√(aᵐ) — denominator = index, numerator = power.
| Exponent form | Radical form | Simplest form |
|---|---|---|
| 2^(3/4) | ∜(2³) | ∜8 |
| 8^(2/3) | ∛(8²) | 4 |
| 2^(5/2) | √(2⁵) | 4√2 |
| 5^(2/3) | ∛(5²) | ∛25 |
| 16^(3/4) | ∜(16³) | 8 |
| 32^(3/5) | ⁵√(32³) | 8 |
| 4^(3/2) | √(4³) | 8 |
| 27^(−2/3) | 1/∛(27²) | 1/9 |
| 12^(1/2) | √12 | 2√3 |
| (−8)^(1/3) | ∛(−8) | −2 |
Common calculator outputs and the exact radicals they come from.
| Decimal | Simplest radical form |
|---|---|
| 20.7846096908 | 12√3 |
| 2.73861278752 | √30/2 |
| 1.41421356 | √2 |
| 0.70710678 | √2/2 |
| 0.866025404 | √3/2 |
| 2.44948974 | √6 |
| 8.48528137 | 6√2 |
| 1.25992105 | ∛2 |
| Radical | Simplest form | Why |
|---|---|---|
| ∜9 | √3 | exponents and index share 2 |
| ⁶√8 | √2 | exponents and index share 3 |
| ∜36 | √6 | exponents and index share 2 |
| ⁶√27 | √3 | exponents and index share 3 |
| ⁶√4 | ∛2 | exponents and index share 2 |
| ⁸√16 | √2 | exponents and index share 4 |
Perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256
Perfect cubes: 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331
Perfect 4th powers: 16, 81, 256, 625, 1296, 2401
Perfect 5th powers: 32, 243, 1024, 3125
Every table above is generated by the same code that runs the calculator. The rules follow the textbook definitions below; the outside and inside parts of a simplified √n match OEIS sequences A000188 and A007913.
Need only square roots? Try the simplify square root calculator. Working with longer expressions? Use the radicals calculator.
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