Square Root Calculator
Use this square root calculator to find √x, a cube root or any root up to the 100th. Besides the decimal value, it simplifies the radical (√72 = 6√2), tells you whether the number is a perfect square or cube, and shows the two whole numbers the root falls between. Negative numbers work too: odd roots give a real answer, even roots an imaginary or complex one.
Updated
Results
- Simplified radical formShown when x is a whole number
- Perfect power?Whether the root is exact
- Between consecutive integersUseful for estimating
- Note
Results are estimates. See the methodology below.
How to use the calculator
- Enter the number (x). It can be a whole number, a decimal or a negative number, up to 10^15 in size.
- Choose the root index (n). Use 2 for a square root, 3 for a cube root, or any whole number up to 100.
- Read the results. The root comes first, then the simplified radical form, whether x is a perfect power, and the integers on either side of the root.
The simplified form is shown when x is a whole number (but not for an even root above 2 of a negative number). The bounds appear when the root is a real number and x is not a perfect power.
How the calculator works
The calculator returns the principal root: for a square root, the non-negative number whose square is x.
- Perfect powers are checked exactly. The number is read digit by digit as you typed it, and an exact integer root is tested. So √144 = 12, √2.25 = 1.5 and ∛0.001 = 0.1 are recognized as exact.
- Other roots are computed as x^(1/n) and shown to 12 significant digits: √72 = 8.48528137424.
- Simplified radical form comes from the prime factorization. Each complete group of n equal prime factors moves outside the radical as one factor; whatever is left stays inside. 72 = 2³ × 3², so √72 = 2 × 3 × √2 = 6√2.
- Negative numbers: an odd index gives a negative real root. An index of 2 gives an imaginary result, i√|x|. An even index above 2 gives the principal complex root, |x|^(1/n) (cos π/n + i sin π/n).
Worked examples
| Input | Root | Simplified | Between |
|---|---|---|---|
| √144 | 12 | 12 | perfect square |
| √72 | 8.48528137424 | 6√2 | 8 and 9 |
| √50 | 7.07106781187 | 5√2 | 7 and 8 |
| √200 | 14.1421356237 | 10√2 | 14 and 15 |
| ∛54 | 3.77976314968 | 3∛2 | 3 and 4 |
| ⁵√96 | 2.49146187923 | 2⁵√3 | 2 and 3 |
1. Simplifying √72 by hand
Look for the largest perfect square that divides 72. That is 36, so √72 = √36 × √2 = 6√2. The calculator does the same thing through prime factors and confirms the decimal value, 8.48528137424, which falls between 8 and 9.
2. Roots of negative numbers
∛(−54) = −3.77976314968, or −3∛2, a real number because (−3∛2)³ = −54. √(−72) has no real value: the result is 8.48528137424i, or 6i√2. ⁴√(−16) gives the complex value 1.41421356237 + 1.41421356237i.
3. A very large number
√999,999,999,999,999 = 31,622,776.6017, simplified to 3√111,111,111,111,111. The root lies between 31,622,776 and 31,622,777.
Reading the result
When perfect power says yes, the root is exact. Otherwise the decimal is rounded, and the simplified radical is the exact answer to use in algebra. The bounds help you estimate without a calculator: √50 is a bit more than 7 because 7² = 49. If you need a set number of decimal places, round the result with the rounding calculator.
Frequently asked questions
Why does the calculator give only the positive square root?
The symbol √ means the principal square root, the non-negative one. The equation x² = 72 still has two solutions, +√72 and −√72. To see both, use the exponent calculator in solve-for-base mode: a^2 = 16 returns ±4.
What is the square root of a negative number?
There is no real square root of a negative number. Using the imaginary unit i, where i² = −1, √(−1) = i and √(−72) = 6i√2. Many calculators and programming languages return an error instead; MDN documents that JavaScript's Math.sqrt returns NaN for x < 0.
How do I simplify a square root?
Split the number into a perfect square times what is left, then take the root of the perfect square. √300 = √100 × √3 = 10√3. For cube roots, look for a perfect cube: ∛54 = ∛27 × ∛2 = 3∛2.
Can the calculator find the square root of a decimal?
Yes. Decimals that are perfect squares are found exactly, such as √2.25 = 1.5. Others, like √0.5 = 0.707106781187, are shown to 12 significant digits. The simplified radical form is only shown for whole numbers.
How do I find a cube root or nth root?
Set the root index to 3 for a cube root, or to any n from 2 to 100. ∛125 = 5 and ∛1,000 = 10. An nth root is the same as raising to the power 1/n, so the exponent calculator gives the same values.
Sources
This calculator is a math tool for learning and checking work. Results that are not exact are rounded to 12 significant digits.