Root Calculator
Calculate square roots, cube roots and any nth root, with simplified radical form and a check of the answer.
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Tool guide
What this tool does
The Root Calculator finds the nth root of any number, including real cube roots of negative numbers, and simplifies integer roots into surd form such as √72 = 6√2. It also lists the square, cube, fourth, fifth, sixth and tenth roots of the same number for quick comparison.
The nth root of x is the number r for which r^n = x. Square roots (n = 2) appear in geometry and statistics, cube roots in volume problems, and higher roots in growth rates: the average yearly growth over 5 years is the fifth root of the total growth factor. Most roots are irrational, so you usually need either a decimal approximation or an exact simplified radical.
This calculator gives both. The decimal value is computed as x^(1/n) and snapped to an exact integer when the input is a perfect power, so √144 shows 12 rather than 11.999999. For whole-number radicands up to a trillion it also extracts every perfect nth-power factor, which turns √72 into 6√2 and ³√54 into 3³√2. Odd roots of negative numbers return the real negative root; even roots of negatives are flagged as having no real value.
How to use Root Calculator
- Enter the radicand, the number under the root sign.
- Enter the root degree n: 2 for a square root, 3 for a cube root, and so on up to 1000.
- Read the highlighted decimal root; the label shows the expression you calculated.
- If the radicand is a whole number with a square (or nth-power) factor, read the simplified radical form.
- Use the Check card to confirm that the root raised to n gives back the radicand, and scroll down for the table of other roots.
Tips for better results
- An even root of a positive number has two real answers, ±r. The calculator reports the principal (positive) root and lists both.
- To find an average growth rate, take the nth root of (final ÷ initial) and subtract 1.
- A radicand that simplifies to a leading coefficient and 1, such as √49 = 7, is a perfect power and has an exact answer.
Common use cases
- Simplifying surds for algebra and trigonometry homework.
- Finding the side of a square or cube from its area or volume.
- Computing geometric means and compound growth rates.
- Checking engineering formulas that involve square or cube roots.
Common mistakes to avoid
- Expecting a real square root of a negative number; it is imaginary (√−9 = 3i).
- Assuming √(a + b) equals √a + √b; roots do not distribute over addition.
- Rounding an irrational root too early in a multi-step calculation, which compounds error.
Frequently asked questions
How do you simplify a square root?
Find the largest perfect square that divides the number and take its root outside: 72 = 36 × 2, so √72 = 6√2. The same idea works for cube roots with perfect cubes.
Can you take the cube root of a negative number?
Yes. Odd roots of negative numbers are real and negative: the cube root of −27 is −3 because (−3)³ = −27.
Why does √(−4) show an imaginary answer?
No real number squared gives a negative result, so the square root of −4 is the imaginary number 2i. Even roots of negatives have no real value.
How accurate are the decimal results?
Results use IEEE 754 double precision, giving about 15 to 16 significant digits. Perfect powers are detected and shown as exact integers.
Feedback & suggestions
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