Enter a number above to calculate its square root.
Square Roots Explained
A square root is the number that, multiplied by itself, gives the original number: √144 = 12 because 12 × 12 = 144. Numbers that are not perfect squares have irrational roots, so √2 = 1.41421356 and the digits never end or repeat. Enter any number above to get its square, cube or nth root, the simplified radical and a check.
Every positive number has two square roots, one positive and one negative, because 12 × 12 and (−12) × (−12) both equal 144. The √ symbol always means the positive one, called the principal root. In exponent form √x = x1/2, the cube root is x1/3, and the nth root is x1/n, which is exactly how this calculator works out every root.
How to Find a Square Root by Hand
1. Bracket it between perfect squares
For √50, note that 7² = 49 and 8² = 64. So the answer is between 7 and 8, and much closer to 7.
2. Improve the guess with the Babylonian method
Take a guess g, divide the number by it, and average the two: new guess = (g + N ÷ g) ÷ 2. Each round roughly doubles the number of correct digits.
| Round | Calculation | New guess |
|---|---|---|
| Start | 7 (from step 1) | 7 |
| 1 | (7 + 50 ÷ 7) ÷ 2 | 7.0714286 |
| 2 | (7.0714286 + 50 ÷ 7.0714286) ÷ 2 | 7.0710678 |
Two rounds already give √50 = 7.0710678, correct to 7 decimal places.
3. Check by squaring
7.0710678 × 7.0710678 = 49.9999999, so the answer is right. Squaring your result is the quickest way to catch a slip.
Simplified Radicals and Decimal Values
To simplify, pull the largest perfect square factor out of the root: √72 = √(36 × 2) = 6√2. A number with no perfect square factor above 1, like 2, 3 or 10, is already in simplest form.
| Number | Simplified | Decimal | Number | Simplified | Decimal |
|---|---|---|---|---|---|
| 2 | √2 | 1.4142 | 24 | 2√6 | 4.8990 |
| 3 | √3 | 1.7321 | 27 | 3√3 | 5.1962 |
| 5 | √5 | 2.2361 | 32 | 4√2 | 5.6569 |
| 6 | √6 | 2.4495 | 45 | 3√5 | 6.7082 |
| 7 | √7 | 2.6458 | 48 | 4√3 | 6.9282 |
| 8 | 2√2 | 2.8284 | 50 | 5√2 | 7.0711 |
| 10 | √10 | 3.1623 | 72 | 6√2 | 8.4853 |
| 12 | 2√3 | 3.4641 | 75 | 5√3 | 8.6603 |
| 18 | 3√2 | 4.2426 | 98 | 7√2 | 9.8995 |
| 20 | 2√5 | 4.4721 | 200 | 10√2 | 14.1421 |
Negative Numbers, Decimals and Common Mistakes
Negative numbers
No real number squared gives a negative, so √(−9) is the imaginary number 3i, where i = √(−1). The calculator shows that form, and √(−72) as 6√2 i. Odd roots are different: the cube root of −8 is simply −2, because (−2)³ = −8. Even roots of 4 or higher of a negative number have no real value.
Zero, one and numbers between 0 and 1
√0 = 0 and √1 = 1. For a number between 0 and 1 the root is bigger than the number itself: √0.25 = 0.5 and √0.01 = 0.1. That is not an error, since multiplying two numbers below 1 always gives something smaller.
Mistakes to avoid
- Splitting a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7. Roots split over multiplication and division, never over addition or subtraction.
- Forgetting the negative root in equations. x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own is just 7.
- Squaring a negative in your head. √(x²) equals the absolute value of x, so √((−5)²) = 5, not −5.
Square roots turn up constantly in geometry. The diagonal of a square with side s is s√2, and the Pythagorean theorem calculator uses √(a² + b²) to find a missing side. For powers and fractional exponents, try the exponent calculator.
Method and sources. The calculator computes the nth root as x1/n with the JavaScript power function and shows up to 10 significant digits. Simplified radicals come from the largest perfect square factor. The by-hand method is the Babylonian (Heron) iteration, a special case of Newton’s method. Every value in the tables above was computed and checked by squaring.
Perfect Squares Reference Table
Click any value to calculate its square root instantly.
Square Root Questions
What is a square root?
A square root of a number x is a value y such that y × y = x. For example, the square root of 25 is 5 because 5 × 5 = 25. Every positive number has two square roots: a positive and a negative one (both 5 and −5 squared give 25). By convention, the "square root" refers to the principal (positive) root, written as √25 = 5.
What is a perfect square?
A perfect square is a number whose square root is a whole integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Non-perfect squares like 2, 3, 5, 7 have irrational square roots: they cannot be expressed as a fraction and their decimal form never ends or repeats. √2 = 1.41421356... continues infinitely.
How do you simplify a square root?
To simplify √72: find the largest perfect square that divides 72. That is 36 (since 36 × 2 = 72). Then: √72 = √(36 × 2) = √36 × √2 = 6√2. The simplified form keeps the perfect square factor outside the radical sign and the remaining factor inside. This calculator shows the simplified form automatically for integer inputs.
What is the square root of 2?
√2 = 1.41421356237... It is irrational, meaning its decimal never ends or repeats. This was one of the first numbers proved irrational, by the ancient Greeks. It appears everywhere in geometry: the diagonal of a 1×1 square has length √2. It also appears in the ISO 216 paper size standard: A4 paper has a ratio of 1:√2 so that folding it in half gives the same proportions.
Can you take the square root of a negative number?
Not in real numbers. The square root of a negative number is imaginary. In complex number math, √(−1) = i (the imaginary unit). So √(−9) = 3i. Imaginary numbers are used in electrical engineering, quantum mechanics, control theory, and signal processing. Enter a negative number above and the square root is shown in this imaginary form. Odd roots of negatives are real, so the cube root of −8 is −2.
What is the cube root?
The cube root of x is the number y such that y × y × y = x. Written as ∛x or x^(1/3). Examples: ∛8 = 2 (2×2×2=8). ∛27 = 3 (3×3×3=27). ∛125 = 5. Unlike square roots, cube roots exist for negative numbers: ∛(−8) = −2 ((−2)×(−2)×(−2) = −8). Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
How does an nth root work?
The nth root of x (written ⁿ√x or x^(1/n)) is the number y such that y^n = x. Square root = 2nd root. Cube root = 3rd root. 4th root of 81 = 3 (3^4 = 81). 5th root of 32 = 2 (2^5 = 32). Fractional exponents express roots: x^(1/n) = ⁿ√x. More generally, x^(m/n) = (ⁿ√x)^m. This is how calculators compute any root using the power function.
What is the square root of 0 and 1?
√0 = 0 (because 0 × 0 = 0). √1 = 1 (because 1 × 1 = 1). Both are perfect squares. √1 is also notable because 1 is the only positive number whose square root equals itself (other than conceptually treating 0 as its own square root). For any n, the nth root of 0 = 0 and the nth root of 1 = 1.
How do you estimate a square root without a calculator?
Find the two perfect squares it falls between. Example: √50. Since 49 = 7² and 64 = 8², √50 is between 7 and 8, closer to 7. Better estimate: 50 is 1/15 of the way from 49 to 64, so √50 ≈ 7 + 1/15 ≈ 7.07. Actual: 7.0711. Another method: the Babylonian method (Newton’s method) repeatedly applies x_new = (x + n/x) / 2 starting from a guess, converging quickly to the answer.
What is the relationship between square roots and exponents?
√x = x^(1/2). This is because x^(1/2) × x^(1/2) = x^(1/2+1/2) = x^1 = x. More generally, ⁿ√x = x^(1/n). Square root laws: √(ab) = √a × √b. √(a/b) = √a / √b. (√a)² = a. These allow simplification of complex radical expressions. Note: √(a+b) ≠ √a + √b, this is a common error.
Why is the square root of a number between 0 and 1 bigger than the number?
Because multiplying two numbers smaller than 1 gives an even smaller result. √0.25 = 0.5, and 0.5 is bigger than 0.25 because 0.5 × 0.5 = 0.25. The same holds for any value between 0 and 1: √0.01 = 0.1 and √0.5 is about 0.7071. For numbers above 1 it works the other way, and the root is smaller than the number.
What is the square root of 3?
√3 = 1.7320508 to 7 decimal places. Like √2 it is irrational, so the decimal never ends or repeats, and 3 has no perfect square factor, so √3 is already in simplest form. It shows up in geometry: an equilateral triangle with sides of 2 has a height of exactly √3.