Simplifying a radical means rewriting it so the number under the radical sign is as small as possible: 3√72 becomes 18√2, ∛−54 becomes −3∛2 and √(5/12) becomes √15/6. Simplified radicals are easier to compare, combine and check, and they are the standard form for exact answers in algebra and geometry. This calculator works with any coefficient and any index, and shows each factor that leaves the radical.
How to use the simplify radicals calculator
- Enter the coefficient, the number already in front of the radical. Leave it blank (or enter 1) if there is none; fractions and negatives are fine.
- Enter the index: 2 for a square root, 3 for a cube root, and so on up to 64.
- Enter the radicand, a whole number or a fraction. Negative radicands work with odd indexes, and with square roots they produce an imaginary result.
- The tape shows the simplified form and its decimal value. The steps and the factor table show how the answer was reached.
The rules behind simplification
The first rule pulls perfect powers out of the radical. The second rationalizes a fraction by multiplying inside the radical until the denominator is a perfect nth power.
A quick way to apply the first rule is the factor table the calculator produces. For each prime, divide its exponent by the index: the quotient tells you how many copies come out, and the remainder tells you how many stay inside.
Worked example: 3√72
Prime factors: 72 = 2³ × 3².
Pairs come out: 2³ contains one pair of 2s with one 2 left over; 3² is one pair of 3s. So 2 and 3 come out, and a single 2 stays inside: √72 = 6√2.
Same idea, faster: the largest perfect square dividing 72 is 36, so √72 = √36 · √2 = 6√2.
Coefficient: 3 × 6√2 = 18√2 ≈ 25.4558441227.
| Prime | Exponent | Groups of 2 (come out) | Left inside |
|---|---|---|---|
| 2 | 3 | 1 | 1 |
| 3 | 2 | 1 | 0 |
Two more examples
- Cube root of a negative: ∛−54. Since 54 = 2 × 3³, the group of three 3s comes out and the sign stays: −3∛2.
- A fraction: √(5/12). Multiply inside by 12/12 to get √60/12. Then √60 = √4 · √15 = 2√15, and 2√15/12 reduces to √15/6.
Common mistakes
- Taking out the wrong factor. √72 is not 2√18 in final form, because 18 still contains the perfect square 9. Always use the largest perfect power, or keep simplifying until no square factor remains.
- Forgetting the coefficient. In 3√72, the 6 that comes out must be multiplied by the 3 already outside.
- Splitting a sum. √(9 + 16) is √25 = 5, not √9 + √16 = 7. The product rule works for multiplication only.
- Leaving a radical in the denominator. 1/√2 is correct but not simplified; the standard form is √2/2.
Combining radicals after simplifying
Simplification often reveals like radicals hidden inside unlike-looking expressions:
√50 + √18 − √8 = 5√2 + 3√2 − 2√2 = 6√2
√12 · √27 = √324 = 18 (multiply first, then simplify)
Radicals with different radicands after simplification, such as √2 and √3, cannot be added.
For decimal values of roots without a coefficient, the square root calculator and the nth root calculator are quicker. The prime factorization calculator shows the factor tree used in the first step.
Frequently asked questions
What makes a radical fully simplified?
Three conditions: no factor of the radicand is a perfect power of the index (other than 1), there is no fraction under the radical, and there is no radical in a denominator. 18√2 meets all three; 3√72 and √(1/2) do not.
How do you simplify a radical with a number in front?
Simplify the radical first, then multiply what comes out by the coefficient. 3√72 = 3 × 6√2 = 18√2.
How do you rationalize a denominator?
Multiply the top and bottom by whatever makes the denominator a perfect power. For √(5/12), multiply inside by 12/12 to get √60/12, then simplify √60 = 2√15 and reduce: 2√15/12 = √15/6.
Can I add radicals after simplifying?
Only like radicals, with the same index and the same radicand, combine. √50 + √18 = 5√2 + 3√2 = 8√2, but √2 + √3 cannot be combined further.