Square root

Calculator for computing a square root

Square root calculator

What is calculated?

This function returns the square root of the given argument (radicand). The argument must be a non-negative real number.

Input values


Result
The result is shown with the selected number of decimal places

Square root info

Properties

Square root:

  • Inverse function of x²
  • Defined for x ≥ 0
  • Result is always non-negative
  • Strictly increasing

Note: For negative or complex numbers a separate root function is available.

Examples
√4 = 2
2² = 4
√9 = 3
3² = 9
√16 = 4
4² = 16
√25 = 5
5² = 25
Advanced functions

For complex or negative numbers a separate root function is available: → Complex square root


Formulas of the square root

Definition
\[\sqrt{x} = x^{1/2}\] Square root as a power
Inverse relation
\[\sqrt{x} = y \Leftrightarrow y^2 = x\] Definition via squaring
Product rule
\[\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\] Root of a product
Quotient rule
\[\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}\] Root of a quotient
Power rule
\[\sqrt{x^a} = x^{a/2}\] Root of a power
Derivative
\[\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}\] Derivative of the square root

Calculation example

Example: calculate √16

Given:

  • Radicand = 16
  • Sought: √16

Calculation:

\[\sqrt{16} = 4\] \[\text{since } 4^2 = 16\]

Result: The square root of 16 is 4.

Geometric example

Example: side length of a square

Problem:

A square has area 25 cm². How long is a side?

Solution:

\[\text{Area} = s^2 = 25\] \[s = \sqrt{25} = 5 \text{ cm}\]

Application: The square root is commonly used in geometry to compute side lengths.

Pythagoras theorem

Example: calculate hypotenuse

Given:

Right triangle with legs a = 3 and b = 4

Hypotenuse c:

\[c^2 = a^2 + b^2 = 3^2 + 4^2 = 25\] \[c = \sqrt{25} = 5\]

Result: The hypotenuse has length 5 units.

Definition and properties

Uniqueness of the solution

In general there exist two different numbers whose squares equal the given number. For example (-3)² = (-3) · (-3) = 9 and 3² = 3 · 3 = 9.

Restriction to non-negative values

When working with real numbers only the principal (non-negative) root is taken. The result is therefore always non-negative.

Important properties
  • Domain: x ≥ 0
  • Range: y ≥ 0
  • Monotonicity: strictly increasing
  • Continuity: continuous on [0, ∞)




More Exp, Log and Root functions

Exponential function base 10x  •  Exponential function base ex  •  Exponential function base xy  •  Modular Exponentiation (PowMod)  •  Logarithm to base 10 (Log)  •  Logarithm to base e (Ln)  •  Logarithm to base a (Loga)  •  Square root (sqrt)  •  Cubic root (cbrt)  •  nth root  •  Hypot  •