Square Root Calculator
Calculate square roots, cube roots, and nth roots
Square Root Calculator
Results
Square Root (√144)
12.000000
Cube Root (∛144)
5.241483
2th Root
12.000000
Squared (144²)
20736.00
Cubed (144³)
2985984.00
144^2
20736.00
Pro Tip: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √144 = 12 because 12 × 12 = 144.
Privacy & Security
Your data is completely private. All calculations are performed locally in your browser—no data is transmitted, stored, or tracked. Your information remains confidential and secure.
What is a Square Root Calculator?
A square root calculator is a powerful mathematical tool that helps you find the root of any number quickly and accurately. Beyond simple square roots, this versatile calculator can compute cube roots, nth roots, and even raise numbers to various powers. Whether you're a student learning about radicals, an engineer working on technical calculations, or anyone who needs to find roots in daily life, this calculator provides instant, accurate results. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 × 4 = 16. This fundamental concept extends to higher-order roots: the cube root of 27 is 3 (because 3 × 3 × 3 = 27), and the fourth root of 81 is 3 (because 3 × 3 × 3 × 3 = 81). Understanding roots is essential for solving quadratic equations, calculating distances in geometry, analyzing scientific data, and numerous engineering applications. Our calculator not only provides the answer but helps you understand the mathematical relationships between numbers and their roots, making it an invaluable learning and practical tool for students, professionals, and anyone working with mathematical calculations.
Key Features
Square Root Calculation
Calculate square roots (√) of any positive number with precision
Cube Root Support
Find cube roots (∛) for volumetric and three-dimensional calculations
Nth Root Calculator
Calculate any root (4th, 5th, 10th, etc.) for advanced mathematical needs
Power Calculator
Raise numbers to any power, including fractional and negative exponents
Decimal Precision
Results displayed with 6 decimal places for maximum accuracy
Instant Results
Real-time calculations as you type with no waiting
Multiple Operations
Calculate roots and powers in the same interface
Free & Private
No registration required, all calculations done in your browser
How to Use the Square Root Calculator
Enter Your Number
Input the number you want to find the root of. For example, enter 144 to find its square root (which is 12).
Select Root Type
Choose the root you want to calculate: square root (2), cube root (3), or any other nth root. The default is square root (2).
View Square Root
See the square root result instantly. For 144, you'll see 12.000000.
Check Cube Root
View the cube root if needed. The calculator shows both square and cube roots simultaneously.
Calculate Nth Root
For custom roots, adjust the 'power' field to calculate any nth root. For example, power=4 calculates the fourth root.
View Powers
See the number raised to various powers (squared, cubed, nth power) for comparison and verification.
Square Root Calculator Tips
- Verify Perfect Squares: Memorize common perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) to quickly recognize when roots will be whole numbers
- Check Your Work: Verify your root by raising it to the corresponding power—the result should equal your original number
- Use for Geometry: Calculate diagonal lengths using the Pythagorean theorem: diagonal = √(side² + side²)
- Estimate First: Develop estimation skills: √50 is between √49 (7) and √64 (8), so it's approximately 7.1
- Convert Fractions: For fractional roots, convert to decimal first: √(1/4) = √0.25 = 0.5
- Round Appropriately: Round your answer to the precision needed for your application—construction needs 2-3 decimals, science may need 6
Frequently Asked Questions
What is a square root and how is it different from other roots?
A square root is a number that, when multiplied by itself, equals the original number. It's the inverse operation of squaring a number. For example, the square root of 25 is 5 because 5 × 5 = 25. The square root symbol is √ and is also called a radical. This differs from other roots in the degree of multiplication: cube roots involve three identical factors (∛27 = 3 because 3 × 3 × 3 = 27), fourth roots involve four factors, and so on. Square roots are the most common type of root you'll encounter in mathematics, used in countless applications from the Pythagorean theorem to standard deviation calculations in statistics. The concept extends to any positive real number, though perfect squares (1, 4, 9, 16, 25, etc.) have whole number roots while most numbers have decimal roots.
Can I calculate the square root of negative numbers?
In real number mathematics, you cannot calculate the square root of a negative number because no real number multiplied by itself produces a negative result. However, in complex number mathematics, the square root of negative numbers is defined using imaginary numbers. The imaginary unit 'i' is defined as √(-1), so √(-4) = 2i, √(-9) = 3i, and so on. Complex numbers have important applications in engineering, physics, and advanced mathematics, particularly in signal processing, quantum mechanics, and electrical engineering. Our calculator works with real numbers and positive values only. If you need to work with complex numbers and negative roots, you would need a specialized calculator that handles imaginary numbers. For most practical applications in everyday calculations, measurement, and basic mathematics, working with positive numbers and their real roots is sufficient.
How accurate are the square root calculations?
Our square root calculator provides results accurate to 6 decimal places, which is more than sufficient for virtually all practical applications. The calculator uses high-precision floating-point arithmetic based on JavaScript's built-in Math functions, which implement IEEE 754 double-precision standards. This means calculations are accurate to approximately 15-17 significant digits internally, though we display 6 decimal places for readability. For most scientific, engineering, and everyday calculations, 6 decimal places provides exceptional accuracy—far beyond what's needed for construction, cooking, financial calculations, or even most scientific work. Only specialized scientific computing or theoretical mathematics might require greater precision. The accuracy is consistent across all root types (square roots, cube roots, nth roots) and power calculations, ensuring reliable results for homework, professional work, or personal calculations.
What are cube roots and when would I use them?
A cube root is a number that, when multiplied by itself three times, equals the original number. The cube root of 27 is 3 because 3 × 3 × 3 = 27, written as ∛27 = 3. Cube roots are essential for three-dimensional calculations involving volume. If you know the volume of a cube and want to find the length of its sides, you calculate the cube root of the volume. For example, a cube with a volume of 64 cubic units has sides of length ∛64 = 4 units. Cube roots appear frequently in physics for inverse cube law calculations, in chemistry for molecular structure analysis, in finance for compound interest calculations over multiple periods, and in computer graphics for 3D modeling and animation. Unlike square roots which only work with positive numbers in real mathematics, cube roots can handle negative numbers because multiplying three negative numbers gives a negative result. For instance, ∛(-8) = -2 because (-2) × (-2) × (-2) = -8.
How do I calculate nth roots for any number?
Calculating an nth root means finding a number that, when multiplied by itself n times, equals your original number. In our calculator, you can set the 'power' field to any positive number to calculate the corresponding nth root. For example, set power to 4 to find fourth roots, 5 for fifth roots, 10 for tenth roots, and so on. The mathematical relationship is: if y = x^(1/n), then y is the nth root of x. Fourth roots are useful in statistics (fourth moment, kurtosis), engineering stress-strain calculations, and signal processing. Fifth roots and higher appear in specialized scientific calculations, polynomial equations, and advanced mathematics. To verify your nth root is correct, you can raise the result to the nth power and confirm it equals your original number. Our calculator shows both the root and power calculations so you can verify the relationship. The nth root becomes increasingly close to 1 as n gets larger (for numbers greater than 1), which is an important mathematical property used in limit calculations and asymptotic analysis.
What's the relationship between roots and exponents?
Roots and exponents are inverse operations—they 'undo' each other. A square root is equivalent to raising a number to the power of 1/2: √x = x^(1/2). Similarly, a cube root equals the 1/3 power: ∛x = x^(1/3), and any nth root equals the 1/n power: ⁿ√x = x^(1/n). This relationship is fundamental in algebra and allows you to solve complex equations by converting between root notation and exponential notation. For example, √16 = 16^(1/2) = 4, and ∛8 = 8^(1/3) = 2. This connection extends to fractional exponents: x^(2/3) means 'cube root of x squared' or equivalently 'square the cube root of x.' Understanding this relationship lets you manipulate equations more flexibly, combine multiple operations efficiently, and solve equations that would be difficult in root notation alone. In calculus, converting roots to fractional exponents makes differentiation and integration much easier. Most scientific calculators and programming languages use this exponential representation internally, even when you enter root symbols.
Why do I get decimal answers instead of whole numbers?
Most numbers don't have exact whole number roots—only perfect powers produce whole number results. Perfect squares like 4, 9, 16, 25, 36 have exact integer square roots (2, 3, 4, 5, 6 respectively), but most numbers like 2, 3, 5, 6, 7, 8, 10 have irrational square roots that continue infinitely without repeating. √2 ≈ 1.414213... and √3 ≈ 1.732050... are famous irrational numbers. Our calculator displays these as decimals with 6 decimal places, providing a very close approximation for practical use. The same applies to cube roots and higher-order roots—only perfect cubes (8, 27, 64) have whole number cube roots. These irrational roots appear constantly in real-world calculations: the diagonal of a unit square is √2, important geometric ratios involve √3 and √5, and the golden ratio involves √5. The decimal approximations are perfectly acceptable and accurate for virtually all applications including construction, science, engineering, and finance.
Can this calculator help me solve equations with radicals?
Yes! Our square root calculator is an excellent tool for solving equations involving radicals (root symbols). If you have an equation like √x = 5, you can find x by squaring both sides (x = 25), then verify your answer by entering 25 into the calculator to confirm √25 = 5. For equations like ∛x = 4, cube both sides (x = 64) and verify with the calculator. The calculator also helps solve more complex radical equations by letting you test values quickly. For instance, if you need to solve √(x + 5) = 7, you can determine x + 5 = 49, so x = 44, then verify by testing √49 = 7. The calculator is particularly useful for checking homework problems, verifying algebraic solutions, or understanding the numerical relationships in radical equations. You can also use it to approximate solutions to equations that don't have nice whole number answers, such as √(2x + 3) = 5.7, by working backward from the decimal result. This makes it an invaluable learning tool for students and a practical verification tool for anyone working with mathematical equations.
Why Use Our Square Root Calculator?
Our square root calculator combines ease of use with powerful mathematical capabilities, making it perfect for students, professionals, and anyone needing quick, accurate root calculations. Unlike basic calculators that only handle square roots, our tool calculates cube roots, nth roots, and powers all in one interface. The real-time calculation feature provides instant feedback as you adjust numbers, helping you understand mathematical relationships better. With 6 decimal place accuracy, clean design, and no registration requirements, it's the fastest way to get reliable root calculations. Whether you're solving homework problems, working on engineering projects, or need quick calculations for daily tasks, our calculator delivers professional results in seconds.