Radical Calculator
A radical like √72 or ∛250 is exact but hard to read, while its decimal (8.485 or 6.300) is readable but rounded. This calculator gives you both: the simplified radical form, such as 6√2 or 5∛2, and the decimal value, for square roots, cube roots, and any nth root. Four tabs cover plain numbers, roots with variables, roots of fractions, and arithmetic between two radicals, and every result comes with the steps that produced it.
Get the simplified radical form and the decimal value for any integer radicand.
Even roots of negative numbers have no real value; the calculator shows the complex form instead. Odd roots of negative numbers are real and negative (for example, the cube root of −8 is −2).
Quick Answer
√72 simplifies to 6√2 (decimal 8.485281) and ∛250 simplifies to 5∛2 (decimal 6.299605). The method is prime factorization: factor the radicand, pull one copy of each prime out of the radical for every full group of n equal factors, and leave the leftover factors inside. Try the pre-filled 72 on the Simplify tab, or ∛(16/54) = 2/3 on the Fraction tab.
Key takeaways
- A radical is in simplest form when no prime factor is left inside with an exponent of n or more.
- The simplified form is exact; the decimal is an approximation cut off at the precision you choose.
- Odd roots of negative numbers are real and negative; even roots of negatives are complex and this tool shows them with i.
- Only like radicals combine: simplify first, then add or subtract the coefficients of matching radicals.
- Quick Answer
- How to use the radical calculator
- How radical simplification works: factor, group, pull out
- Worked examples, verified in the calculator
- Radicals with variables: exponents split by the index
- Fraction radicands: rationalizing works for every index
- Adding, subtracting, multiplying, and dividing radicals
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the radical calculator
Pick the tab that matches your expression, enter the values, and click Calculate. The Simplify tab takes an index and an integer radicand; With variables adds a variable string such as x^4 y^3; Fraction takes a numerator and denominator; Arithmetic takes two radicals and an operation.
Three limits are worth knowing before you start. Radicands on the Simplify, Fraction, and Arithmetic tabs must be whole numbers, so a decimal like 7.5 is rejected with an explanation rather than silently rounded. The Simplify tab caps the radicand at 1 trillion, which keeps the prime factorization instant. And variables are assumed positive, so √(x²) returns x rather than |x|, the standard textbook convention that OpenStax states for principal square roots of variable expressions.
How radical simplification works: factor, group, pull out
Simplification works by prime factorization. The calculator breaks the radicand into prime factors, groups them into sets matching the root index, pulls each complete group outside the radical as a single copy of that prime, and leaves the remainder inside.
Take √72. Factor 72 into primes: 72 = 2³ × 3². For a square root, group the factors into pairs: one pair of 2s comes out as a single 2, the pair of 3s comes out as a 3, and one 2 is left inside. So √72 = 2 × 3 × √2 = 6√2. For a cube root the groups are triples: ∛250 = ∛(5³ × 2) = 5∛2. In exponent form the same move is ⁿ√x = x^(1/n), the identity behind the exponent calculator; if you want to practice rewriting radicals as rational exponents and back, the convert to exponential form calculator drills exactly that step.
72 = 2³ × 3²
Pairs for √: (2·2) (3·3) and one spare 2
Each pair leaves as one copy: 2 and 3
6√2, with the spare 2 inside
Worked examples, verified in the calculator
Every row below was run through the tool above, so you can re-enter any of them and land on the same numbers. The Type column is the label the Simplify tab reports.
| Expression | Simplified form | Decimal | Where |
|---|---|---|---|
| √72 | 6√2 | 8.485281 | Simplify (Irrational) |
| ∛250 | 5∛2 | 6.299605 | Simplify (Irrational) |
| ∛(−8) | −2 | -2.000000 | Simplify (Perfect cube) |
| √(−8) | 2√2i | 2.828427i | Simplify (Complex) |
| √1,000,000 | 1000 | 1000.0000 at 4 places | Simplify (Perfect square) |
| √(9/16) | 3/4 | 0.750000 | Fraction |
| ∛(16/54) | 2/3 | 0.666667 | Fraction |
| √(72x⁴y³) | 6x²y√2y | Numeric part 8.485281 | With variables |
| 2√3 + 5√3 | 7√3 | 12.124356 | Arithmetic |
| (3√2) ÷ (2√3) | (√6)/2 | 1.224745 | Arithmetic |
Radicals with variables: exponents split by the index
The With variables tab applies the same grouping rule to exponents. For each variable, divide its exponent by the index: the quotient becomes the exponent outside the radical and the remainder stays inside.
So √(72x⁴y³) splits three ways at once. The number 72 gives 6 outside and 2 inside, as before. The x⁴ gives x² outside with nothing left over, because 4 ÷ 2 = 2 exactly. The y³ gives y outside and y inside, because 3 ÷ 2 = 1 remainder 1. Combined: 6x²y√2y. Multi-letter names work too, and repeated variables merge, so entering x^2 x^3 is read as x^5. One caution: because variables are assumed positive, the tool never prints absolute-value bars. If your variables can be negative in the original problem, an even root of an even power needs |x| in a fully rigorous answer, a caveat OpenStax flags when it defines a simplified radical expression.
Fraction radicands: rationalizing works for every index
The Fraction tab simplifies ⁿ√(p/q) by simplifying the numerator and denominator roots separately, then rationalizing the denominator with the general rule ⁿ√(b/d) = ⁿ√(b × d^(n−1))/d.
That general rule matters. The shortcut √(p/q) = √(pq)/q is only valid for square roots, because it quietly uses d^(n−1) with n = 2. Watch the difference on a cube root: ∛(16/54). Simplifying each part gives 2∛2 over 3∛2; rationalizing the denominator root (∛(2 × 2²) = 2) and reducing 4/6 lands on exactly 2/3, which matches the decimal 0.666667. A tool that applies the square-root shortcut to a cube root returns a form whose value disagrees with its own decimal, so check any fraction result by comparing the simplified form against the decimal beside it. If your fraction starts life as decimals, convert them to integers first with the decimal to fraction calculator, since this tab works with whole numbers only.
Adding, subtracting, multiplying, and dividing radicals
The Arithmetic tab simplifies both radicals first, then combines them by the standard rules: like radicals add and subtract by coefficient, products merge under one radical, and quotients rationalize the denominator.
Like means same index and same simplified radicand. That is why √18 + √8 works out so cleanly: the tab simplifies them to 3√2 and 2√2, and the coefficients combine to 5√2. Radicals that simplify to different radicands, like √2 and √3, are left as a sum, because √2 + √3 is not √5. Both radicals share the one index you pick in the Root index (both) field, so a cube-root problem such as ∛16 + ∛54 is handled as cube roots throughout and gives 5∛2. Division follows the quotient property: (3√2) ÷ (2√3) = (√6)/2, with no radical left in the denominator. If you only ever work with cube roots, the dedicated cube root calculator covers that single index in more depth.
What competing calculators miss
Most radical tools do one job. The gaps below are the ones that send people to a second or third site.
- Decimal-only answers. Several nth-root calculators return a rounded decimal and nothing else. A decimal cannot be carried into the next algebra step; the exact form 6√2 can.
- Positive integers only. Widely used simplifying tools state that even roots of negative numbers are imaginary and unsupported, and stop there. This tool returns the complex form, 2√2i for √(−8), and handles odd roots of negatives as the real negative numbers they are.
- No variables, fractions, or arithmetic. The big tutoring-style root calculators simplify one numeric radicand well, but homework rarely stops there. Variable radicands, fraction radicands with rationalized denominators, and adding or dividing two radicals live on separate pages elsewhere; here they are tabs in one tool.
- Worked examples you cannot trust. One popular nth-root page works the fourth root of 196 as 4, which fails its own check: 4⁴ = 256, not 196. The correct result is 4√196 ≈ 3.741657, and it is the answer this calculator gives. Every example here was reproduced in the tool itself.
- Fraction shortcuts that break past square roots. Tools that rationalize with √(pq)/q for every index produce cube-root forms that contradict their own decimals. The general d^(n−1) rule used here keeps the exact form and the decimal in agreement at any index.
Frequently asked questions
The radicand is the number under the radical sign. In √72, the radicand is 72. The small number showing which root to take, the 2 in a square root or the 3 in a cube root, is called the index.
Not as a real number. Even roots of negatives are complex: √(−8) = 2√2 i. Odd roots of negatives are fine and stay negative: ∛(−8) = −2. The Simplify tab shows the complex form for even roots instead of refusing the question.
A radical is in simplest form when the radicand has no perfect-power factors left for its index: no perfect squares under a square root, no perfect cubes under a cube root. So √72 must become 6√2, and ∛54 must become 3∛2.
The simplified radical is exact: 6√2 is precisely √72. The decimal 8.485281 is an approximation that gets cut off wherever you stop. Exact form is what algebra and geometry proofs need; decimals are what measurements need.
Simplify first, then combine only matching radicals: √18 + √8 = 3√2 + 2√2 = 5√2. Radicals with different radicands, like √2 + √3, cannot be combined into a single radical. The Arithmetic tab does the simplify-then-combine steps for you.
Take out full pairs: 72 = 36 × 2 gives 6√2; x^4 gives x^2; y^3 gives y√y. Combined: 6x^2y√2y. The With variables tab does this automatically, and the same divide-the-exponent rule works at any index.
3/4. Rationalize: √(9/16) = √144/16 = 12/16 = 3/4. The Fraction tab handles any integer fraction and reduces the result.
2/3. Simplify each part (∛16 = 2∛2 and ∛54 = 3∛2), rationalize the denominator with the general rule ⁿ√(b/d) = ⁿ√(b × d^(n−1))/d, and reduce 4/6 to 2/3. The square-root shortcut √(pq)/q does not work at index 3.
Yes: divide the coefficients and the radicands, then rationalize the denominator. (3√2)/(2√3) = √6/2. The Arithmetic tab shows the steps.
Simplification is prime factorization, and factorization is defined for whole numbers. A decimal radicand has no exact simplified radical form to find, so the tool asks for an integer instead of rounding your input silently. For a decimal approximation of a decimal root, raise it to the power 1/n on the exponent calculator instead.
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References
- OpenStax, Rice University. 9.1 Simplify and Use Square Roots, Elementary Algebra 2e (principal root, square roots of negative numbers not real, variables assumed non-negative). Accessed October 24, 2026.
- OpenStax, Rice University. 9.2 Simplify Square Roots, Elementary Algebra 2e (product and quotient properties, perfect-square factors, simplifying variable expressions). Accessed October 24, 2026.
- OpenStax, Rice University. 8.2 Simplify Radical Expressions, Intermediate Algebra (definition of a simplified radical expression for any index n). Accessed October 24, 2026.
- OpenStax, Rice University. 9.8 Rational Exponents, Elementary Algebra 2e (radicals rewritten as rational exponents, ⁿ√x = x^(1/n)). Accessed October 24, 2026.
- Weisstein, Eric W. Radical, Wolfram MathWorld (radicand, index and vinculum terminology; roots as fractional exponents). Accessed October 24, 2026.
- Weisstein, Eric W. nth Root, Wolfram MathWorld (principal root, real odd roots of negative numbers, roots of unity background). Accessed October 24, 2026.