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 approximate. 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.
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, and leaves the remainder inside. Every step of the factorization is shown, so the result is checkable, not just believable.
Radical Calculator
Simplified radical form plus decimal value for square, cube, and nth roots.
| Expression | — |
| Decimal value | — |
| Type | — |
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).
How radical simplification works
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.
Worked examples
√72 simplifies to 6√2 (≈ 8.485281). ∛250 simplifies to 5∛2 (≈ 6.299605). ∛(−8) = −2, because odd roots of negative numbers are real and negative. √(−8) has no real value; the exact complex form is 2√2 i (≈ 2.828427i).
Radicals with variables
The With variables tab simplifies roots like sqrt(72x^4y^3). The rule is the same as for numbers: each full group of n equal factors comes out of an nth root. Here 72 = 6^2 x 2, x^4 gives x^2 outside, and y^3 gives y outside with y left inside, so the answer is 6x^2y sqrt(2y). Type variables as letters with ^exponent, separated by spaces (x^4 y^3 z). Variables are assumed positive and stay symbolic; the decimal shown covers the numeric part only.
Fraction radicands
The Fraction tab simplifies roots of fractions by rationalizing the denominator: sqrt(p/q) = sqrt(pq)/q. So sqrt(9/16) = sqrt(144)/16 = 12/16 = 3/4, and sqrt(2/3) = sqrt(6)/3. The outside fraction is always reduced: sqrt(18/50) = sqrt(900)/50 = 30/50 = 3/5.
Radical arithmetic
The Arithmetic tab adds, subtracts, multiplies, and divides radicals. Like radicals combine by their coefficients (2 sqrt(3) + 5 sqrt(3) = 7 sqrt(3)); unlike radicals stay as a simplified sum. Products multiply inside one radical (2 sqrt(3) x 3 sqrt(2) = 6 sqrt(6)); quotients rationalize the denominator ((3 sqrt(2))/(2 sqrt(3)) = sqrt(6)/2).
Worked examples
- sqrt(72x^4y^3): 72 = 2^3 x 3^2, so 6 comes out with sqrt(2) inside; x^4 gives x^2; y^3 gives y with y inside. Result: 6x^2y sqrt(2y).
- cbrt(16/54): rationalize to cbrt(864)/54; 864 = 2^5 x 3^3, so 6 comes out with cbrt(4) inside; reduce 6/54 to 1/9. Result: (cbrt(4))/9.
- (3 sqrt(2))/(2 sqrt(3)): (3/2) x sqrt(2/3) = (3/2) x (sqrt(6)/3) = (3 sqrt(6))/6 = sqrt(6)/2.
Enter these into the matching tabs to verify each result.
Frequently asked questions
What is a radicand?
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.
Can you take the square root of a negative number?
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.
What does it mean for a radical to be simplified?
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.
Why do calculators show two answers, exact and decimal?
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.
How do I add radicals like √18 + √8?
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.
How do you simplify sqrt(72x^4y^3)?
Take out full pairs: 72 = 36 x 2 gives 6 sqrt(2); x^4 gives x^2; y^3 gives y sqrt(y). Combined: 6x^2y sqrt(2y). The With variables tab does this automatically.
What is sqrt(9/16) simplified?
3/4. Rationalize: sqrt(9/16) = sqrt(144)/16 = 12/16 = 3/4. The Fraction tab handles any integer fraction and reduces the result.
Can you divide radicals?
Yes: divide the coefficients and the radicands, then rationalize the denominator. (3 sqrt(2))/(2 sqrt(3)) = sqrt(6)/2. The Arithmetic tab shows the steps.