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The square root of a nonnegative number is the nonnegative value that squares to give that number. For 81, the principal square root is 9 because 9 × 9 = 81. The radical symbol √ means this principal root.
This page calculates a real square root and rounds the result to seven decimal places. It does not display complex roots for negative inputs. Use it to check a square-root calculation, recover a square’s side length from its area or compare an irrational root with nearby whole numbers.
Zero is allowed: its square root is zero. A negative input has no real square root and should not be treated as a valid real-number example on this page.
| Quantity | Explanation |
|---|---|
| x | The nonnegative radicand, or number under the square-root symbol. |
| y | The principal square root. It is nonnegative even when an equation involving y² also has a negative solution. |
| Units | If x is an area in square centimeters, y is a length in centimeters. Units must be interpreted alongside the numerical operation. |
For x = 144, the result is 12 because 12² = 144. This is an exact whole-number root, so 144 is a perfect square. For x = 150, the root lies between 12 and 13 because 144 < 150 < 169.
For x = 2, the display is approximately 1.4142136. Squaring that rounded value gives a number close to 2, but not necessarily exactly 2. The exact value is written √2 and has a nonterminating, nonrepeating decimal expansion.
A square with area 64 square meters has side length √64 = 8 meters. This works because a square’s area is side × side. A rectangle with the same area can have many different pairs of side lengths, so its area alone does not determine either side through a square root.
Since 0.3 × 0.3 = 0.09, √0.09 = 0.3. The root of a positive number below one is larger than the number itself. This is a useful scale check and prevents the assumption that every square-root operation must make a number smaller.
Square roots appear when reversing a squared quantity, finding geometric lengths and manipulating algebraic expressions. The calculator evaluates the numerical root, while the problem determines whether it represents a distance, a ratio or something else.
The expression √49 has the single value 7. The equation y² = 49 has two real solutions: 7 and −7. Both statements are correct because they ask different questions. In a length problem the negative solution is usually excluded by the physical meaning, not because (−7)² fails to equal 49.
An exercise may prefer √12 = 2√3 rather than a decimal approximation. This form displays the numerical value; it does not simplify radicals symbolically. Keep exact radical expressions in algebraic work where later terms may cancel, and use decimals when the requested output is numerical.
The square-root symbol denotes the principal nonnegative root. A plus-or-minus sign is needed when solving an equation such as y² = x with x positive, because both signs can satisfy that equation.
A decimal can have an exact rational square root: √0.25 = 0.5. The term perfect-square integer usually refers specifically to a nonnegative integer that is the square of an integer.
There is no real square root of a negative number. Complex-number calculations use the imaginary unit i, but this page’s real square-root calculation does not provide that type of result.
Find neighboring squares. Since 8² = 64 and 9² = 81, √70 lies between 8 and 9. Further trial values narrow the interval and provide a reasonableness check on a computed answer.
For nonnegative real a and b, √(ab) = √a × √b. This lets you simplify √72 as √(36 × 2) = 6√2. The nonnegative-domain condition matters; extending the rule carelessly to negative factors can give contradictions. This product rule also does not turn into a corresponding rule for addition inside a root.
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