Root Calculator
Square, cube and any nth root to 30 digits, with simplified radical form like √72 = 6√2, exact roots of perfect powers and complex roots of negatives.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Root Calculator?
Pick square, cube, fourth or any nth root up to the 1000th, type the number under the root, and the calculator gives the decimal value to 30 significant digits together with the simplified radical form your textbook wants: √72 = 6√2, ∛(−54) = −3∛2, √(1/2) = √2/2.
When the number is a perfect power the answer is exact, even for numbers far beyond what a phone calculator can hold, such as √12345678987654321 = 111111111. When it is not, the steps show the prime factorisation, how equal primes are grouped and pulled out of the radical, and which two whole numbers the root sits between.
Even roots of negative numbers have no real value, so the calculator gives the principal complex root instead, and for indices up to 12 it lists all n complex roots, which are spaced evenly around a circle.
- Square, cube, fourth and nth roots up to n = 1000
- Simplified radical form with rationalised denominators
- Exact answers for perfect powers of any size
- 30 correct significant digits for irrational roots
- Principal complex root and all n complex roots
- Prime-factor steps and integer bounds
How to use the Root Calculator
- Choose the root: square, cube, fourth, or nth root and type n.
- Type the number under the root. Whole numbers, decimals, fractions such as 3/4, negatives and E notation all work.
- Read the decimal value and the simplified radical. Perfect powers are marked exact.
- Open the steps to see the prime factorisation and how the factors leave the radical.
- For negative numbers under an even root, read the principal complex root and the full list of complex roots.
Worked examples
Simplify √72
Input
√72
Output
6√2 ≈ 8.48528137423857
72 = 2³ × 3². One pair of 2s and one pair of 3s come out as 2 × 3 = 6; a single 2 stays inside.
Cube root of a negative
Input
∛(−54)
Output
−3∛2 ≈ −3.77976314968462
Odd roots keep the sign. 54 = 2 × 3³, so a 3 comes out and 2 stays under the cube root.
Rationalising
Input
√(1/2)
Output
√2/2 ≈ 0.707106781186548
Multiply inside by 2/2 to get √(2/4) = √2 / 2, which is the same as 1/√2 without a root in the denominator.
Even root of a negative
Input
√(−12)
Output
2i√3
There is no real answer; the principal value is i times √12, and √12 simplifies to 2√3.
Background and insight
The nth root of x is the number r with rⁿ = x. Written as a power it is x^(1/n), which is why roots and fractional exponents follow the same rules. For a positive x there is exactly one positive real nth root, called the principal root, and that is what the radical sign means. For a negative x, odd roots are real and negative (∛−27 = −3) while even roots are not real at all, because any real number raised to an even power is zero or positive.
Simplifying a radical means writing it as k·ⁿ√m with m as small as possible. Factor the number into primes; every complete group of n identical primes leaves the radical as a single factor. 72 = 2³ × 3², so √72 takes out one 2 and one 3 and leaves a single 2 inside: 6√2. A fraction is rationalised first by multiplying top and bottom so the denominator becomes a perfect power, which is why √(2/3) is written √6/3.
Irrational roots are computed here with 80-digit decimal arithmetic and shown to 30 digits, so the last digit you see is correctly rounded. For comparison, ordinary double-precision floating point, used by most calculators and spreadsheets, carries about 15 to 17 significant digits.
Common use cases
Algebra homework
Check simplified radical answers and see the factor grouping, which is the step most worksheets mark.
Geometry and trigonometry
Keep exact side lengths such as 5√2 for the diagonal of a 5-unit square instead of a rounded decimal.
Engineering and finance formulas
Take nth roots for growth rates, where the annual rate is the nth root of total growth minus 1, or for scaling laws with cube and fourth roots.
Checking large numbers
Find whether a big integer is a perfect square or cube without rounding errors.
Tips and limitations
- The radical sign means the principal (positive) root. Both 6 and −6 square to 36, but √36 is 6.
- For a fractional power such as x^(2/3), use the Exponent Calculator, which handles roots and powers together.
- Very large integers (over 60 digits) still get an exact perfect-power test, but the simplified radical needs a full factorisation and is skipped.
- Complex roots are rounded to 12 digits in the table; the principal real root is shown to 30.
Frequently asked questions
How do you simplify a square root?
Factor the number into primes, take one prime out for every pair, and leave the unpaired primes inside. 72 = 2 × 2 × 2 × 3 × 3, so the pairs give 2 × 3 = 6 outside and one 2 stays in: √72 = 6√2.
What is the cube root of a negative number?
A real negative number. (−3)³ = −27, so ∛(−27) = −3. Odd roots of negatives are always real; only even roots of negatives need complex numbers.
What is the square root of a negative number?
It is not a real number. Using i, where i² = −1, √(−x) = i√x for positive x, so √(−12) = i√12 = 2i√3.
How many nth roots does a number have?
Every non-zero number has exactly n complex nth roots, evenly spaced around a circle in the complex plane. At most two of them are real. The calculator lists all of them for n up to 12.
Why do I need to rationalise the denominator?
It is a convention that makes answers comparable and easier to add: √2/2 and 1/√2 are equal, but the first form is the standard way to write it. Multiply top and bottom by whatever makes the denominator a perfect power.
How accurate are the decimals?
Roots are computed with 80 significant digits and displayed to 30, so every digit shown is correct. Perfect powers are detected with whole-number arithmetic and shown exactly.
Is a root the same as a fractional exponent?
Yes. The nth root of x equals x^(1/n), and the nth root of x to the power m equals x^(m/n). That is why 8^(2/3) is (∛8)² = 4.
Is there an in-depth guide to the Root Calculator?
Yes. “Square Roots, Cube Roots and nth Roots: Simplifying Radicals Properly” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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