A square root calculator that also simplifies radicals
Type any number into this square root calculator to get √x instantly, rounded to anything from 0 to 50 decimal places. Switch to cube roots or any nth root with one tap. For whole numbers it also gives the simplest radical form — √72 = 6√2, ∛54 = 3∛2 — and says whether the number is a perfect square or cube.
Unlike an ordinary calculator, the digits are computed with exact integer arithmetic, so √2 to 50 places is correct to the last digit. The working shows the prime factorisation behind each simplified radical, a few rounds of Newton’s method converging on the answer, and a table of the nearest perfect squares — handy for estimating roots in your head before an exam.
How to use the Square Root Calculator
- For a printable list, choose Roots table, set the range (1–30, 1–50 or any up to 500 rows), the decimal places and the columns, then print or download the PDF or PNG.
- Choose Square root, Cube root or nth root. For an nth root, enter the index n (for example 4 or 5).
- Type the number. Decimals, negatives and scientific notation such as 1.5e8 are accepted.
- Pick how many decimal places you want, from 0 to 50.
- Read the root in the result panel; the caption shows the simplest radical form when it can be simplified.
- Use the tiles and the working to see the factorisation, the perfect-power check and the nearby perfect squares, then copy the result.
Square Root Calculator features
Exact digits up to 50 places
Roots are calculated with big-integer arithmetic rather than floating point, so long answers have no rounding drift.
Simplest radical form
Extracts every complete group of prime factors: √180 = 6√5, ∛250 = 5∛2, ⁴√48 = 2⁴√3.
Perfect square and cube check
Tells you straight away whether the root is a whole number or a terminating decimal such as √2.25 = 1.5.
Negative numbers handled properly
Odd roots of negatives are real (∛−27 = −3); even roots give imaginary or complex results (√−16 = 4i).
Learn the method
Prime factorisation, Newton’s method iterations and a table of neighbouring perfect powers for estimation.
Printable roots tables
Squares, cubes, square roots and cube roots for any range, whole-number roots shaded, on A4 or Letter as PDF, PNG or print.
How to simplify a square root
Write the number as a product of primes, then take one factor outside the root for every pair. For √72: 72 = 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, so 2 × 3 = 6 comes out and a single 2 is left inside: √72 = 6√2. For cube roots you need groups of three: 54 = 2 × 3 × 3 × 3, so ∛54 = 3∛2.
A quicker route is to spot the largest perfect square factor: 72 = 36 × 2, and √36 = 6. The prime method always works, even when the largest square factor is hard to see.
Estimating a square root without a calculator
Find the two perfect squares either side of your number. 72 lies between 64 (8²) and 81 (9²), so √72 is between 8 and 9 — a little closer to 8.5. Then improve the guess with the Babylonian method: average the guess with the number divided by the guess. (8.5 + 72 ÷ 8.5) ÷ 2 = 8.4853, already correct to four decimal places. This is Newton’s method for x² − S = 0, and it is what the working panel shows.
| n | n² | n³ | √n (6 d.p.) |
|---|---|---|---|
| 2 | 4 | 8 | 1.414214 |
| 3 | 9 | 27 | 1.732051 |
| 5 | 25 | 125 | 2.236068 |
| 7 | 49 | 343 | 2.645751 |
| 10 | 100 | 1,000 | 3.162278 |
| 12 | 144 | 1,728 | 3.464102 |
| 15 | 225 | 3,375 | 3.872983 |
| 20 | 400 | 8,000 | 4.472136 |
Common mistakes with roots
- Splitting a sum. √(9 + 16) is √25 = 5, not √9 + √16 = 7. Roots split over products (√(4 × 9) = √4 × √9) but never over sums.
- Forgetting the absolute value. √(x²) = |x|, so √((−5)²) = 5, not −5.
- Losing the negative solution. x² = 49 has two solutions, x = 7 and x = −7, even though the symbol √49 means only the principal (positive) root, 7.
- Square roots of negatives. √−16 is not −4; it is 4i, an imaginary number. Cube roots are different: ∛−8 = −2 because (−2)³ = −8.
Square Root Calculator FAQ
What is the cube root of 1728?
12, because 12 × 12 × 12 = 1,728 — choose Cube root and type 1728, or find it in the n³ column of the roots table at row 12. Other perfect cubes to 30³ appear in the 1–30 table.
How do I find the square root of a number?
Enter the number in the square root calculator and the answer appears instantly. By hand, find the nearest perfect squares to estimate, then refine with the Babylonian method: average your guess with the number divided by your guess, and repeat.
What is the simplest radical form of a square root?
It is the root written as a whole number times the smallest possible root. √50 = √(25 × 2) = 5√2. The calculator finds it from the prime factorisation for any whole number up to 10¹⁵.
Can you take the square root of a negative number?
Not within the real numbers. The result is imaginary: √−25 = 5i, where i² = −1. Cube roots and other odd roots of negative numbers are real, so ∛−125 = −5.
How do I calculate a cube root or nth root?
Choose Cube root or nth root, enter n and the number. The nth root of x is the number that gives x when raised to the power n, so ⁵√32 = 2 because 2⁵ = 32. It is the same as raising x to the power 1/n, which the Exponent Calculator can also do.
How do I know if a number is a perfect square?
A whole number is a perfect square when every prime in its factorisation has an even exponent: 144 = 2⁴ × 3², so √144 = 12. The “Perfect square?” tile checks this exactly, even for very large numbers.
Why does my calculator give a slightly different last digit?
Most calculators use floating-point numbers with about 15 significant digits and round the final digit. This tool computes digits exactly and rounds half up at the place you choose, so its digits are correct as shown. Square roots also appear in the Pythagorean Theorem Calculator when finding a hypotenuse.