Square Root Calculator
Calculate square roots, simplify radical expressions, and find perfect squares
Calculate Square Roots
Negative numbers will result in complex numbers (with i)
Choose precision for decimal approximation
Square Root Result
Step-by-Step Solution:
- 1. √0 = 0
Perfect Squares Reference
Square Root Properties
Definition
√x = y means y² = x
Multiplication
√(a×b) = √a × √b
Division
√(a/b) = √a / √b
Common Irrational Square Roots
Quick Tips
√x² = |x| for all real numbers
Perfect squares have exact integer roots
√0 = 0 and √1 = 1
Negative numbers have imaginary roots
Simplify by extracting perfect square factors
Understanding Square Roots
What is a Square Root?
A square root of a number x is a value that, when multiplied by itself, gives x. For example, the square root of 9 is 3 because 3 × 3 = 9. Every positive number has two square roots: one positive and one negative.
Perfect Squares
Perfect squares are numbers that are the product of an integer multiplied by itself. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. These numbers have exact integer square roots.
Applications
- •Geometry: Finding side lengths of squares
- •Physics: Calculating distances and velocities
- •Statistics: Standard deviation calculations
- •Engineering: Signal processing and optimization
Simplifying Square Roots
Example: Simplify √72
Step 1: Factor 72 = 36 × 2
Step 2: √72 = √(36 × 2) = √36 × √2
Step 3: √72 = 6√2
Example: Simplify √50
Step 1: Factor 50 = 25 × 2
Step 2: √50 = √(25 × 2) = √25 × √2
Step 3: √50 = 5√2
Square Root Symbol History
The √ symbol originated from the letter "r" (for radix, meaning root in Latin) and was first used in the 16th century. The horizontal bar (vinculum) was added later to clearly show what's under the radical.