Math
Simplify Radicals Calculator
Enter a radicand, pick the root index, and optionally add a coefficient to simplify expressions like √72, ∛54, or 5√72.
The panel shows the prime factorization, which factors come out of the radical in groups, the simplest radical form, the decimal value, and the equivalent rational-exponent form. Negative radicands are handled for odd roots (real) and even roots (imaginary).
The number multiplying the radical. Use 1 for a plain root.
A whole number up to 1 trillion. Negatives allowed.
Try an example
Result
Simplest radical form
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- Decimal
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- Prime factorization
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- Rational exponent form
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What the simplify radicals calculator solves
A A root expression written with the √ symbol, such as a square root. is in simplest form when no perfect square (or perfect cube for cube roots) is left under the root sign. √72 is not simplified because 72 = 36 × 2 and 36 is a perfect square; the simplest form is 6√2. This calculator factors the radicand into A whole number above 1 whose only factors are 1 and itself., pulls out every complete group, and multiplies any The number multiplying a variable, like the 3 in 3x. you entered.
The prime-factor method
- Write the radicand as a product of primes: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
- For a square root, circle pairs of identical primes; for a cube root, groups of three.
- Each complete group contributes one copy of that prime outside the radical: one pair of 2s and one pair of 3s give 2 × 3 = 6.
- Leftover primes stay inside: one 2 remains, so √72 = 6√2.
- Multiply by any coefficient in front: 5√72 = 5 × 6√2 = 30√2.
| Expression | Factorization | Simplest form | Decimal |
|---|---|---|---|
| √72 | 2³ × 3² | 6√2 | 8.4853 |
| √50 | 2 × 5² | 5√2 | 7.0711 |
| ∛54 | 2 × 3³ | 3∛2 | 3.7798 |
| ∜48 | 2⁴ × 3 | 2∜3 | 2.6321 |
| √-18 | -(2 × 3²) | 3i√2 | not real |
Negative radicands
Odd roots of negative numbers are real: ∛-54 = -3∛2 because (-3∛2)³ = -54. Even roots of negatives are not real; the calculator writes them with the imaginary unit i, so √-18 = 3i√2.
How to use it
- Enter the radicand (the number under the root sign).
- Choose square, cube, fourth, or fifth root.
- Enter a coefficient if the expression has one in front, like the 5 in 5√72; leave it at 1 otherwise.
- Read the simplest radical form and decimal, then check the prime-factor table to see which factors came out and which stayed inside.
How to read the answer
The number outside the radical is the product of all the complete groups of primes, and the number inside is the product of the leftovers. If the radicand becomes 1, the original number was a perfect power and the root is a plain integer. The decimal value is the same for the original and simplified forms; simplifying only changes how the exact value is written.
Common mistakes and edge cases
- Pulling out a prime that appears only once: √12 = 2√3, not 4√3 or 2√6.
- Using the wrong group size: for a cube root you need three identical primes to pull out one, so ∛16 = 2∛2, not 4∛2.
- Forgetting to multiply the coefficient you started with: 5√72 = 30√2, not 6√2.
- Adding radicals with different radicands: √2 + √3 is not √5.
- Treating √-16 as -4. The real square root does not exist; the answer is 4i.
Worked examples
Simplify √72
72 = 2³ × 3², so one pair of 2s and one pair of 3s come out.
Simplest radical form
6√2
With a coefficient: 5√72
5 × 6√2 = 30√2
Simplest radical form
30√2
Cube root ∛54
54 = 2 × 3³
Simplest radical form
3∛2
Negative cube root ∛-54
Odd roots of negatives are real.
Simplest radical form
-3∛2
Perfect square √144
The root comes out completely.
Simplest radical form
12
Already simplest: √30
No prime repeats, so nothing comes out.
Simplest radical form
√30
Imaginary: √-18
Even root of a negative.
Simplest radical form
3i√2
Decimal radicand
Whole numbers only.
Simplest radical form
Error
Frequently asked questions
What is simplest radical form?+
A radical is simplified when the radicand has no perfect-square factor (for square roots) or perfect-cube factor (for cube roots) other than 1, and no radical sits in a denominator. √72 in simplest radical form is 6√2.
How do you simplify the square root of 72?+
Factor 72 = 36 × 2. Since 36 is a perfect square, √72 = √36 × √2 = 6√2, which is about 8.4853.
How do you simplify a cube root?+
Look for groups of three identical prime factors instead of pairs. ∛54 = ∛(27 × 2) = 3∛2 because 27 = 3³.
What do I do with the number in front of the radical?+
Multiply it by whatever comes out of the radical. 5√72 = 5 × 6√2 = 30√2.
Can you simplify the square root of a negative number?+
Not as a real number. Write √-18 = √-1 × √18 = i × 3√2 = 3i√2. Cube roots of negatives are fine: ∛-8 = -2.
Is √30 already simplified?+
Yes. 30 = 2 × 3 × 5 has no repeated prime, so no perfect square divides it and nothing can come out.
How do I write a radical as an exponent?+
The k-th root of b is b^(1/k): √2 = 2^(1/2) and ∛5 = 5^(1/3). A coefficient stays as a multiplier, so 6√2 = 6 × 2^(1/2).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026