Square Root Calculator
Enter a number and get its square root (or cube root, or any root up to the 100th) to as many decimal places as you choose, with the simplest radical form, a perfect-square check and the working. The digits are computed exactly with whole-number arithmetic, not floating point. Free, no sign-up.
A whole number, decimal or fraction: 72, 0.5 or 2/3.
2 for a square root, 3 for a cube root, up to 100.
Results
≈ 8.4852813742
Principal root
6√2
Simplest radical form
No
Perfect square?
The working
- 72 = 2³ × 3²
- Every pair of equal factors leaves the root as a single factor, so 6 comes out and 2 stays inside.
- √72 = 6√2
Between two perfect powers
72 lies between 8² = 64 and 9² = 81, so √72 is between 8 and 9.
Heron’s method
Start from a guess, then replace it by the average of the guess and x divided by the guess. The number of correct digits roughly doubles each step.
| Step | Estimate | Estimate squared |
|---|---|---|
| 0 | 8 | 64 |
| 1 | 8.5 | 72.25 |
| 2 | 8.48529411765 | 72.000216263 |
| 3 | 8.48528137425 | 72.0000000002 |
| 4 | 8.48528137424 | 72 |
| 5 | 8.48528137424 | 72 |
How to Use the Square Root Calculator
- 1. Enter the number under the root. A whole number, a decimal (0.5) or a fraction (2/3). Negative numbers are accepted and handled as described below.
- 2. Choose the index. 2 gives a square root, 3 a cube root, and any whole number up to 100 gives that root.
- 3. Pick the precision. The decimal is computed digit by digit to between 2 and 50 places and rounded correctly. If the root is exact, the calculator says so and shows no padding zeros.
- 4. Read the forms. You get the decimal, the simplest radical form (such as 6√2), whether the number is a perfect power, and the working that gets there.
What a Square Root Is
The square root of x is the non-negative number whose square is x, written √x. Because 8 × 8 = 64, √64 = 8; and because 8.485… squared is very close to 72, √72 ≈ 8.4852813742. A perfect square is the square of a whole number — 1, 4, 9, 16, 25 and so on — and only perfect squares have whole-number roots. Every other positive whole number has an irrational square root: a decimal that never ends and never repeats.
The same idea gives the cube root ∛x, the number whose cube is x, and the n-th root ⁿ√x, the number whose n-th power is x. Odd roots exist for every real number (∛−27 = −3). A square root of a negative number is not real: the principal value is imaginary, √(−16) = 4i, and the calculator reports it that way.
| n | n² | n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1000 |
| 11 | 121 | 1331 |
| 12 | 144 | 1728 |
Simplest Radical Form
A radical is in simplest form when no factor of the number under the root appears often enough to be taken out, and there is no root in a denominator. To simplify √72, factor it: 72 = 2³ × 3². Each pair of equal factors leaves the root as one factor, so a 3 and a 2 come out and a single 2 stays in: √72 = 3 × 2 × √2 = 6√2. For other roots, groups of n equal factors come out: ∛54 = ∛(2 × 3³) = 3∛2.
A fraction or decimal is handled the same way, with primes in the denominator counted as negative powers. √(1/2) becomes √2/2, which is the rationalised form teachers ask for, and √0.5 gives the same answer. The calculator factors the number under the root exactly, so the simplified form is never an approximation.
Worked Example: √72
- Between which perfect squares? 8² = 64 and 9² = 81, so √72 lies between 8 and 9.
- Simplest radical form: 72 = 2³ × 3², so √72 = 6√2.
- Decimal: √72 ≈ 8.4852813742 (10 places), and 6 × √2 = 6 × 1.41421356237 = 8.48528137422 agrees.
- Heron’s method from a guess of 8: the average of 8 and 72 ÷ 8 = 9 is 8.5, then 8.4852941, then 8.4852814 — after three steps the estimate matches the true root to nine decimal places.
Why “√x ≥ 0”
Both 8 and −8 square to 64, but √64 means only the non-negative one, 8. That is the principal square root. When you solve x² = 64 you write ±√64 to get both.
Common Mistakes
- Splitting sums. √(a + b) is not √a + √b: √(9 + 16) = 5, but √9 + √16 = 7. Roots split over products and quotients only.
- Stopping the simplification early. √48 = 2√12 is correct but incomplete; 12 still holds a square factor, so √48 = 4√3.
- Forgetting the denominator. 1/√2 is usually rewritten as √2/2 so that no root is left in the denominator.
- Taking an even root of a negative number and expecting a real answer.
Frequently Asked Questions
- How do you simplify a square root?
- Factor the number into primes, then take one factor out of the root for every pair of equal primes. √72 = √(2 × 2 × 2 × 3 × 3) = 2 × 3 × √2 = 6√2. If no prime appears twice, the root is already in simplest form, as √30 is.
- How can I tell whether a number is a perfect square?
- Take its square root: a perfect square has a whole-number root. Without a calculator, a perfect square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8, and its prime factorization has only even exponents.
- Is the square root of 2 a fraction?
- No. √2 is irrational: it cannot be written as a ratio of two whole numbers, and its decimal 1.41421356… never ends or repeats. The calculator shows as many digits as you ask for, each one exact.
- What is the square root of a negative number?
- There is no real number whose square is negative, so the square root of a negative number is not real. The principal value is imaginary: √(−16) = 4i, where i is defined by i² = −1.
- Why does the calculator say 8.4852813742 with an approximately sign?
- √72 is irrational, so no finite decimal equals it exactly; the value shown is correctly rounded to the places you chose. When a root is exact, as √0.25 = 0.5 is, the calculator shows the exact value without the sign.
- How do you find a square root by hand?
- Estimate it between two perfect squares, then refine with Heron’s method: average the guess with the number divided by the guess, and repeat. Each round roughly doubles the number of correct digits.