Square Root Calculator

Find the square root, cube root, or any nth root instantly.

√ Square Root Calculator
Number
Root Type

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

Square Root: √x = x^(1/2)  |  (√x)² = x
Example: √144 = 12    (12² = 144 ✓)
Cube Root: ∛x = x^(1/3)  |  (∛x)³ = x
Example: ∛27 = 3    (3³ = 27 ✓)
nth Root: ⁿ√x = x^(1/n)  |  (ⁿ√x)ⁿ = x
n = root degree (2 = square, 3 = cube, 4 = fourth, etc.)
x = the radicand (number under the radical)
Example: ⁴√81 = 81^(1/4) = 3    (3⁴ = 81 ✓)
Note: For negative x, only odd roots are real: ∛(−8) = −2

How to Use This Calculator

Steps to Calculate a Root

1
Enter your number in the Number field. This can be any positive number, or a negative number for odd roots.
2
Select the root type: Square Root, Cube Root, 4th Root, or Custom nth Root. For Custom, enter the value of n.
3
Click Calculate to see the result, 10-decimal precision, verification (result^n), and exponent form.

Example Calculations

Example 1 - Square Root of a Non-Perfect Square

Find the square root of 200

1
200 = 4 × 50 = 4 × 25 × 2 = 100 × 2
2
√200 = √(100 × 2) = √100 × √2 = 10√2
3
10 × 1.41421356... = 14.142135...
√200 = 10√2 ≈ 14.1421
Try this example →

Example 2 - Cube Root

Find the cube root of 512

1
Try 8: 8³ = 8 × 8 × 8 = 512 ✓
∛512 = 8 (perfect cube)
Try this example →

Example 3 - nth Root Application

What is the 5th root of 100,000?

1
100,000 = 10⁵, so ⁵√100,000 = 10^(5/5) = 10^1 = 10
⁵√100,000 = 10 (since 10⁵ = 100,000 ✓)
Try this example →

Frequently Asked Questions

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).
Tags: Square Root Sqrt Cube Root Nth Root Radical Power Exponent

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📌 Quick Tips

💡The square root of a perfect square (1, 4, 9, 16, 25...) is always a whole number. Any other positive number has an irrational square root.
💡√(a × b) = √a × √b. You can simplify radicals by factoring out perfect squares. For example, √48 = √(16×3) = 4√3.
💡Negative numbers do not have real square roots. The square root of −1 is the imaginary number i, used in complex number theory.