Square Root Calculator: Find √x Instantly
Our square root calculator estimates the square root of any positive number you want. Just enter the chosen number and read the results. Everything is calculated quickly and automatically! With this tool, you can also estimate the square of the desired number and use it to find perfect squares.
Square Root Symbol √
Square roots were already known in antiquity. The earliest clay tablet with the correct value of √2 = 1.41421 up to 5 decimal places comes from Babylonia (1800 BCE – 1600 BCE). Many other documents show that square roots were also used by the ancient Egyptians, Indians, Greeks, and Chinese. However, the origin of the root symbol √ is still largely speculative.
Many scholars believe that square roots originate from the letter "r" — the first letter of the Latin word radix meaning root. According to another theory, the square root symbol was taken from the Arabic letter ج placed in its original form ﺟ in the word جذر (root). The "bar" over the numbers inside the symbol (√‾) is known as vinculum in Latin, meaning bond.
Why Are Square Roots Called This Way?
The choice of the name "root" becomes more evident if we recast the equation x = ⁿ√a as xⁿ = a. The number x is called a root or radical because it is the hidden base of a. Geometrically, the square root √x is the side length of a square of area x — hence the name. Likewise, the cube root ∛x is the edge length of a cube of volume x.
Square Root Definition
In mathematics, the square root of a given number x is every number y whose square y² = y × y yields the original number x. The square root formula can be expressed as:
y = ±√x ⟺ y² = x
Each positive real number always has two distinct square roots — one positive, one negative. By contrast, 0 has only one root — itself. The symbol √x always denotes the non-negative root of x. There is also a fractional power notation: √x = x1/2 = x0.5.
How to Find the Square Root
The best answer: use this square root calculator! But it helps to remember a few basic perfect square roots:
- √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5
- √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
- √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15
For non-perfect squares, you can estimate by trial and error. For example, to find √52: you know √49 = 7 and √64 = 8, so √52 is between 7 and 8. Since 52 is closer to 49, try 7.2 → 7.2² = 51.84, close enough. Alternatively, simplify: √52 = 2√13 ≈ 2 × 3.61 = 7.22.
How to Simplify Square Roots
To simplify √n, find the largest perfect square factor of n and extract it:
- √12 = √(4 × 3) = 2√3 ≈ 3.46
- √50 = √(25 × 2) = 5√2 ≈ 7.07
- √72 = √(36 × 2) = 6√2 ≈ 8.49
- √200 = √(100 × 2) = 10√2 ≈ 14.14
Operations with Square Roots
Adding and Subtracting
You can only add or subtract square roots with the same radicand (the number under the √ symbol):
- 2√3 + 5√3 = 7√3 ✓
- √2 + √3 ≠ √5 ✗ (cannot combine different radicands)
- √8 + √2 = 2√2 + √2 = 3√2 ✓ (after simplification)
Multiplying Square Roots
√a × √b = √(a × b)
- √3 × √12 = √36 = 6
- 2√5 × 3√5 = 6 × 5 = 30
Dividing Square Roots
√a / √b = √(a/b)
- √50 / √2 = √25 = 5
- √18 / √2 = √9 = 3
Square Roots of Powers and Fractions
For powers: √(x^n) = x^(n/2). For fractions: √(a/b) = √a / √b.
- √(x⁴) = x²
- √(1/4) = 1/2
- √(9/16) = 3/4
Square Root in Geometry
Metric System
If a square has an area of A m², its side length is √A meters. Example: a square room of 25 m² has sides of √25 = 5 m.
US / Imperial System
If a square has an area of A ft², its side length is √A feet. Example: a 144 ft² room has sides of √144 = 12 ft (about 3.66 m).
Derivative of the Square Root
The derivative of √x with respect to x is: d/dx(√x) = 1 / (2√x). This function is defined for all x > 0.
Square Root of a Negative Number
The square root of a negative number is not a real number — it requires complex numbers. We define the imaginary unit i such that i² = −1. Then:
- √(−1) = i
- √(−4) = 2i
- √(−9) = 3i
FAQs
What is the square root of 2?
√2 ≈ 1.41421356... It is an irrational number — its decimal expansion never ends and never repeats.
Is every number the square root of something?
Yes! For any non-negative real number x, x = (x²)^(1/2). There is always a number whose square equals x².
What is a perfect square?
A perfect square is a positive integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Can I calculate the square root without a calculator?
Yes — use trial and error, simplification with prime factors, or the long-division method for manual calculation. But our calculator does it in milliseconds!