Square Root Calculator
How to use this square root calculator
- Enter any non-negative number in the input field. Whole numbers, decimals, and large numbers are all accepted.
- The square root appears instantly below the input with up to 10 decimal places of precision.
- Check whether your number is a perfect square and read the full step-by-step solution above.
- Click Copy to grab the exact result for use in another calculation or application.
Square root formula and rules
Core formula
The square root formula is sqrt(x) = x^(1/2). Raising any non-negative number to the power of one-half gives its principal (positive) square root. This is exactly how the calculator works: it evaluates x^0.5 using the browser's native floating-point engine, which provides up to 15 significant digits of accuracy.
Inverse relationship
(sqrt(x))^2 = x for x greater than or equal to 0, and sqrt(x^2) = |x| for all real x. The absolute value matters: squaring removes the sign, so the square root always returns a non-negative value. Applying both operations in sequence returns the original number (or its absolute value for negative inputs).
Product rule -- the key to simplification
sqrt(a x b) = sqrt(a) x sqrt(b) for a, b greater than or equal to 0. This is the essential rule for simplifying radical expressions. To simplify sqrt(50): 50 = 25 x 2, so sqrt(50) = sqrt(25) x sqrt(2) = 5sqrt(2) approximately 7.071. Always pull out the largest perfect square factor of the radicand in one step to reach the simplest form directly.
Quotient rule
sqrt(a / b) = sqrt(a) / sqrt(b) for a greater than or equal to 0, b greater than 0. Radicals distribute over division just as they do over multiplication. Example: sqrt(25/4) = sqrt(25) / sqrt(4) = 5/2 = 2.5. This rule is especially useful when both numerator and denominator are perfect squares.
Adding and subtracting square roots
You can only combine square roots with the same radicand -- they behave like like terms: 3sqrt(2) + 5sqrt(2) = 8sqrt(2). But sqrt(2) + sqrt(3) cannot be simplified further. This rule is tested constantly in algebra, and it is the reason factoring out perfect squares first is so important -- it often reveals matching radicands that can then be added.
Quick reference
| Rule | Formula | Example |
|---|---|---|
| Core formula | sqrt(x) = x^(1/2) | sqrt(81) = 9 |
| Inverse | (sqrt(x))^2 = x | (sqrt(5))^2 = 5 |
| Product rule | sqrt(a*b) = sqrt(a)*sqrt(b) | sqrt(50) = 5sqrt(2) |
| Quotient rule | sqrt(a/b) = sqrt(a)/sqrt(b) | sqrt(9/4) = 1.5 |
| Like terms | n*sqrt(x)+m*sqrt(x) = (n+m)*sqrt(x) | 3sqrt(7)+2sqrt(7)=5sqrt(7) |
What is a square root?
The square root of a number x is the value y such that y x y = x. Written as sqrt(x) in radical notation or x^(1/2) using exponents, it is the mathematical inverse of squaring. If 9 squared equals 81, then the square root of 81 equals 9. This inverse relationship is fundamental to algebra, geometry, and virtually every area of applied science and engineering.
Every positive real number has two square roots -- a positive one and a negative one. Both 5 squared and (-5) squared equal 25, so the two square roots of 25 are 5 and -5. By convention, the principal square root is always the positive one, so sqrt(25) = 5. When you need both solutions (such as in quadratic equations), write +/-sqrt(25) = +/-5.
The square root of 0 is 0, the only number with just one square root. The square root of a negative number is imaginary -- sqrt(-1) = i, the imaginary unit that underpins complex number theory, electrical engineering, and quantum physics. This calculator handles only non-negative real inputs.
- sqrt(4) = 2 because 2 x 2 = 4 (perfect square)
- sqrt(9) = 3 because 3 x 3 = 9 (perfect square)
- sqrt(2) approximately 1.41421 -- irrational, non-terminating decimal
- sqrt(0.25) = 0.5 -- decimals work too: 0.5 x 0.5 = 0.25
- sqrt(100) = 10 and sqrt(144) = 12
Interpreting your square root result
Perfect square -- exact whole-number result
When the result is flagged as a perfect square, the answer is an exact integer with zero rounding. The verified equation (e.g., 12 x 12 = 144) appears in the explanation. Perfect square roots come up most often in geometry (Pythagorean triples), algebra (factoring expressions), and standardized test math.
Non-perfect square -- irrational decimal
Most numbers are not perfect squares. Their square roots are irrational -- the decimal never terminates or repeats. The calculator shows 10 significant decimal places, which exceeds the precision needed for any engineering, scientific, or financial application. The verification step confirms accuracy by squaring the rounded result back toward the original.
Decimal and fraction radicands
Square roots work on any non-negative decimal. sqrt(0.5) = 1/sqrt(2) approximately 0.70711. sqrt(2.25) = 1.5 exactly, because 2.25 = 9/4 and sqrt(9/4) = 3/2. Enter any decimal in the calculator above and read the precise result immediately.
Real-world applications of square roots
The Pythagorean theorem -- geometry's most famous square root
The hypotenuse of a right triangle is c = sqrt(a^2 + b^2). For legs of 3 m and 4 m: c = sqrt(9 + 16) = sqrt(25) = 5 m. The same formula calculates the straight-line distance between any two points in 2D space, making it fundamental to navigation, construction, architecture, and computer graphics every single day.
Statistics -- standard deviation
Standard deviation is sigma = sqrt(variance) = sqrt(Sum(x - mu)^2 / n). The square root brings variance -- which is measured in squared units -- back to the original data units, making it interpretable. A dataset with variance 25 kg squared has a standard deviation of 5 kg, which you can directly compare to individual measurements. This is why standard deviation is far more commonly reported than variance in research and analytics.
Physics -- velocity and kinetic energy
The kinetic energy formula KE = (1/2)mv^2 rearranges to v = sqrt(2KE / m). Square roots emerge whenever you invert a quadratic relationship -- any formula where a quantity scales with the square of another variable. Escape velocity, wave propagation speed, and resonance frequencies all depend on square roots in their underlying equations.
Electrical engineering -- RMS voltage
In AC circuits, RMS voltage = V_peak / sqrt(2) approximately 0.7071 x V_peak. A standard 120V wall outlet has a peak voltage of about 170V, and 170 / sqrt(2) approximately equals 120V RMS. The sqrt(2) factor comes from integrating a squared sine wave over one full cycle -- a foundational result in every electrical engineering curriculum.
Finance -- portfolio volatility scales with sqrt(time)
Expected portfolio volatility grows with the square root of time. If a stock has annualized volatility of 20%, its daily volatility is approximately 20% / sqrt(252) approximately 1.26%, where 252 is the number of trading days per year. This sqrt-of-time rule comes from the statistical properties of random walks, the mathematical foundation of most financial pricing models including Black-Scholes.
Perfect squares reference table
Memorizing these 15 values gives you instant mental estimation for any square root:
| n | n squared (perfect square) | Square root |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
How to estimate a square root by hand
Knowing the 15 perfect squares above makes mental estimation straightforward. Here is the step-by-step technique:
- Find the two nearest perfect squares. For sqrt(75): 8 squared = 64 and 9 squared = 81. So sqrt(75) lies between 8 and 9.
- Interpolate to get a decimal estimate. 75 is (75-64) / (81-64) = 11/17 approximately 0.65 of the way from 64 to 81. So sqrt(75) approximately 8.65. The exact answer is 8.6603 -- this estimate is within 0.03%.
- Refine with one Newton step (optional). New guess = (guess + n/guess) / 2. With 8.65: (8.65 + 75/8.65) / 2 = (8.65 + 8.671) / 2 approximately 8.66 -- extremely close.
- Verify by squaring. 8.66 squared approximately 74.99 -- accurate to 4 significant figures.
Quick shortcut: if a number is just above a perfect square, the root is the integer plus 1/(2 x integer). sqrt(50) is just above sqrt(49) = 7, so sqrt(50) approximately 7 + 1/14 approximately 7.071. Exact to 3 decimal places.
Quick mental square root tips
Use these patterns to reduce mental arithmetic when a calculator is not available:
- Powers of 10: sqrt(100) = 10, sqrt(10000) = 100. Move the decimal point and halve the zeros.
- Ratio shortcut: sqrt(a/b) = sqrt(a) / sqrt(b). sqrt(4/9) = 2/3 instantly.
- Estimate by bracketing: sqrt(50) is between sqrt(49)=7 and sqrt(64)=8, closer to 7.
- Newton iteration: One step of (guess + n/guess)/2 halves the error in your estimate.
- Simplify first: sqrt(200) = sqrt(100 x 2) = 10*sqrt(2) approximately 14.14. Much easier than estimating sqrt(200) directly.
Common square root mistakes to avoid
sqrt(a + b) is not equal to sqrt(a) + sqrt(b)
This is the single most common radical error. sqrt(9 + 16) = sqrt(25) = 5, but sqrt(9) + sqrt(16) = 3 + 4 = 7. The product rule works (sqrt(a x b) = sqrt(a) x sqrt(b)), but there is no sum rule for radicals. Never split a square root over addition or subtraction.
Forgetting the plus-or-minus sign when solving equations
When solving x^2 = 25, the answer is x = +/-5, not just x = 5. The square root operation returns only the positive principal root. When used to solve equations, both the positive and negative roots must be written. Dropping the negative root leads to missing solutions in physics, quadratic equations, and optimization problems.
Confusing sqrt(x^2) = |x| and (sqrt(x))^2 = x
sqrt((-5)^2) = sqrt(25) = 5, not -5. The absolute value is required because squaring removes the sign. By contrast, (sqrt(5))^2 = 5 -- this one does not need absolute value since sqrt(5) is already non-negative. Know which form you are working with before applying the inverse identity.
Rounding mid-calculation
If sqrt(2) appears in a multi-step formula, keep the full precision (1.41421356...) until the final step. Rounding to 1.41 early and then multiplying by a large number compounds the error significantly. Use this calculator to get the precise value, then carry it unrounded through your calculation.