Square Root Calculator

Calculate square roots and simplify radicals to their simplest form

Calculate Square Root

Enter a number to find its square root and simplified radical form

Number

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. Square roots are fundamental in mathematics, appearing in geometry, algebra, physics, and many real-world applications.

Perfect Squares

Perfect squares are numbers whose square roots are whole numbers. Here are the first 15 perfect squares:

√1

√4

√9

√16

√25

√36

√49

√64

√81

√100

√121

√144

√169

√196

√225

Simplifying Square Roots

How to Simplify

Find the prime factorization of the number
Group factors into pairs of identical numbers
Move each pair outside the radical as a single number
Multiply numbers outside the radical together
Multiply remaining numbers inside the radical together

Example: Simplify √72

1. Prime factors: 72 = 2 × 2 × 2 × 3 × 3

2. Group pairs: (2 × 2) × (3 × 3) × 2

3. Move pairs out: 2 × 3 = 6 outside, 2 inside

4. Result: 6√2

Common Applications

Geometry:Finding side lengths, diagonals, and distances
Physics:Calculating velocity, acceleration, and energy
Statistics:Standard deviation and variance calculations
Engineering:Structural analysis and signal processing
Finance:Risk assessment and portfolio optimization

Key Properties

√(a × b)= √a × √b
√(a / b)= √a / √b
√a²= |a| (absolute value)
(√a)²= a
√1= 1
√0= 0

Calculation Tips

•Square roots of negative numbers are not real numbers (they're imaginary)
•Every positive number has two square roots: one positive and one negative
•The √ symbol always refers to the principal (positive) square root
•Memorizing perfect squares up to 225 helps with quick mental calculations
•Use estimation: √50 is between √49 (7) and √64 (8), closer to 7

Related Calculators

Percentage Calculator

Calculate percentages and percentage changes

Percentage Increase Calculator

Calculate percentage increase

Percentage Decrease Calculator

Calculate percentage decrease

Percentage Difference Calculator

Calculate percentage difference

Fraction Calculator

Calculate with fractions

Ratio Calculator

Calculate ratios and proportions