Square Root Calculator
Enter any number to get its square root, cube root or nth root as an exact simplified radical and as a decimal to the precision you choose, with the prime factorisation shown step by step.
Step-by-step working
Quick examples
How to use the Square Root Calculator
- Type the number (the radicand). Decimals, negatives and scientific notation like 2e6 are accepted.
- Pick the root: square root, cube root, or nth root and type the index n.
- Choose the precision for the decimal answer, from 2 to 50 decimal places.
- Press Calculate (results also update as you type) to see the decimal value, the simplified radical and whether the number is a perfect power.
- Read the steps to see how the prime factors are grouped to simplify the radical.
How to simplify a radical (worked examples)
To simplify the nth root of a whole number, split it into prime factors, take every group of n equal factors outside the root as a single factor, and leave the leftovers inside.
- √72: 72 = 2³ × 3² = (2² × 3²) × 2. The pairs 2² and 3² come out as 2 × 3 = 6, so √72 = 6√2 ≈ 8.4853.
- ∛54: 54 = 2 × 3³. The group 3³ comes out as 3, so ∛54 = 3∛2 ≈ 3.7798.
- √0.5: 0.5 = 1/2 = 2/4, and √4 = 2, so √0.5 = √2/2 ≈ 0.7071. This is called rationalising the denominator.
- ∛−27: odd roots keep the sign, and 3³ = 27, so ∛−27 = −3.
- √−50: 50 = 5² × 2, so √−50 = 5i√2 ≈ 7.0711i, an imaginary number.
Root rules and formulas
| Rule | Formula | Example |
|---|---|---|
| Root as a power | ⁿ√x = x^(1/n) | ∛8 = 8^(1/3) = 2 |
| Product rule | ⁿ√(ab) = ⁿ√a × ⁿ√b | √72 = √36 × √2 = 6√2 |
| Quotient rule | ⁿ√(a/b) = ⁿ√a / ⁿ√b | √(9/16) = 3/4 |
| Root of a power | ⁿ√(xᵐ) = x^(m/n) | √(5⁶) = 5³ = 125 |
| Odd root of a negative | ⁿ√(−x) = −ⁿ√x (n odd) | ⁵√−32 = −2 |
| Square root of a negative | √(−x) = i√x | √−9 = 3i |
Perfect squares and cubes from 1 to 20
| n | n² | n³ | n | n² | n³ |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 11 | 121 | 1,331 |
| 2 | 4 | 8 | 12 | 144 | 1,728 |
| 3 | 9 | 27 | 13 | 169 | 2,197 |
| 4 | 16 | 64 | 14 | 196 | 2,744 |
| 5 | 25 | 125 | 15 | 225 | 3,375 |
| 6 | 36 | 216 | 16 | 256 | 4,096 |
| 7 | 49 | 343 | 17 | 289 | 4,913 |
| 8 | 64 | 512 | 18 | 324 | 5,832 |
| 9 | 81 | 729 | 19 | 361 | 6,859 |
| 10 | 100 | 1,000 | 20 | 400 | 8,000 |
Common irrational roots: √2 ≈ 1.414214, √3 ≈ 1.732051, √5 ≈ 2.236068, √10 ≈ 3.162278, ∛2 ≈ 1.259921, ∛3 ≈ 1.442250.
Frequently asked questions
How do you simplify a square root?
Write the number as a product of prime factors, take out every pair of equal factors and leave the rest under the root. For example 72 = 2 × 2 × 2 × 3 × 3, so √72 = 2 × 3 × √2 = 6√2.
How do you find a cube root?
A cube root is the number that multiplied by itself three times gives the original number. To simplify, take out every group of three equal prime factors: 54 = 2 × 3 × 3 × 3, so ∛54 = 3∛2 ≈ 3.7798.
Can you take the square root of a negative number?
Not in the real numbers, because any real number squared is zero or positive. The answer is imaginary: √−16 = 4i, where i = √−1. Odd roots of negative numbers are real, for example ∛−27 = −3.
What is an nth root?
The nth root of x is the number r with rⁿ = x. The square root is n = 2 and the cube root is n = 3. It is the same as raising x to the power 1/n, so ⁿ√x = x^(1/n).
How do I know if a number is a perfect square?
A whole number is a perfect square when every prime in its factorisation appears an even number of times, for example 144 = 2⁴ × 3². The calculator tells you whether your number is a perfect square, cube or nth power and shows the nearest ones.
How accurate are the decimal answers?
The decimals are computed with exact integer arithmetic rather than ordinary floating point, so every digit shown is correct and the last digit is rounded. You can choose up to 50 decimal places.
What is the square root of 2?
√2 is irrational, so its decimal never ends or repeats. To 20 decimal places it is 1.41421356237309504880.