Square Root Calculator
Find the square root, cube root, or any nth root instantly.
📊 What is a Square Root?
The square root of a number is the value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol √. For example, √25 = 5 because 5 × 5 = 25. Square roots are one of the most frequently used mathematical operations across algebra, geometry, physics, engineering, and everyday problem-solving.
Square roots appear naturally whenever area is involved. If a square field has an area of 625 square metres, then its side length is √625 = 25 metres. Similarly, the distance formula in coordinate geometry (the Pythagorean theorem) requires taking a square root: distance = √((x₂−x₁)² + (y₂−y₁)²).
A perfect square is a number whose square root is an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. For all other positive integers, the square root is an irrational number - it cannot be expressed as a fraction and its decimal expansion continues forever without repeating. For example, √2 = 1.41421356… and √3 = 1.73205080…
The cube root (∛) finds the number that, multiplied by itself three times, gives the original. The nth root generalises this: the nth root of x is the number that raised to the power n gives x. Mathematically, the nth root of x equals x^(1/n), a fractional exponent. This calculator handles all of these cases, including odd roots of negative numbers (which are real).
📐 Formulas
📖 How to Use This Calculator
Steps to Calculate a Root
💡 Example Calculations
Example 1 - Square Root of a Non-Perfect Square
Find the square root of 200
Example 2 - Cube Root
Example 3 - nth Root Application
What is the 5th root of 100,000?
Frequently Asked Questions
🔗 Related Calculators
What is a square root?
The square root of a number n is a value x such that x² = n. For example, the square root of 25 is 5, because 5² = 25. Every positive number has two square roots (positive and negative), but by convention the square root symbol (√) refers to the positive root.
How do you calculate the square root without a calculator?
The most common manual method is the Babylonian (or Newton's) method: start with an initial guess g, then repeatedly improve it using g = (g + n/g) / 2. After a few iterations, g converges to √n. For perfect squares, memorise the table: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10.
What is the cube root?
The cube root of n is the value x such that x³ = n. For example, the cube root of 27 is 3, because 3³ = 27. Unlike square roots, cube roots of negative numbers are real - the cube root of −8 is −2.
What is an nth root?
The nth root of a number n is x such that xⁿ = n. The square root is the 2nd root, the cube root is the 3rd root, and so on. The nth root of x is mathematically equivalent to x^(1/n), a fractional exponent.
Can you take the square root of a negative number?
Not in the real number system. The square root of a negative number is an imaginary number. For example, √(−9) = 3i, where i is the imaginary unit defined as √(−1). This is the domain of complex numbers, used extensively in electrical engineering and physics.
How do you calculate a square root by hand?
The simplest method is the digit-by-digit (long division) method or estimation by perfect squares. For estimation: find the nearest perfect square below and above your number. Example: to estimate sqrt(50): sqrt(49) = 7 and sqrt(64) = 8. So sqrt(50) is between 7 and 8, closer to 7. Refine: 7.07^2 = 49.98 (close enough). The exact value is 7.0710.....
What is the square root of a negative number?
The square root of a negative number is an imaginary number, written using the imaginary unit i, where i = sqrt(-1). Example: sqrt(-9) = sqrt(9) x sqrt(-1) = 3i. Imaginary numbers are part of the complex number system and are fundamental in electrical engineering, quantum physics, and signal processing. They are not real numbers and cannot be plotted on a standard number line.
What are the perfect squares up to 100?
Perfect squares are the squares of integers. From 1 to 100: 1 (1^2), 4 (2^2), 9 (3^2), 16 (4^2), 25 (5^2), 36 (6^2), 49 (7^2), 64 (8^2), 81 (9^2), 100 (10^2). Memorising these speeds up mental math and helps with simplifying radical expressions. For example, to simplify sqrt(72): 72 = 36 x 2, so sqrt(72) = sqrt(36) x sqrt(2) = 6 sqrt(2).