Square Root Calculator

How to Use This Square Root Calculator

1. Enter the number: Input the number (radicand) you want to find the root of. This can be any positive or negative number.
2. Select the root index: Enter 2 for square root (√), 3 for cube root (∛), or any positive integer for nth root.
3. Calculate: Click "Calculate Root" to get the result with detailed step-by-step information.
4. View steps: See the calculation breakdown including the radical notation, decimal result, and verification.

What is a Square Root?

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. The square root symbol (√) is called a radical sign, and the number under it is called the radicand.

Square roots are fundamental in mathematics, appearing in geometry (Pythagorean theorem), algebra (quadratic equations), physics (wave equations), and countless real-world applications from calculating distances to analyzing data distributions.

Understanding Nth Roots

An nth root generalizes the concept of square roots. The nth root of a number x is a value that, when raised to the power n, equals x. For example:

• Square root (n=2): √16 = 4 because 4² = 16
• Cube root (n=3): ∛27 = 3 because 3³ = 27
• Fourth root (n=4): ⁴√81 = 3 because 3⁴ = 81

Properties of Square Roots and Roots

Product property: √(a × b) = √a × √b. For example, √36 = √(4 × 9) = √4 × √9 = 2 × 3 = 6.

Quotient property: √(a ÷ b) = √a ÷ √b. For example, √(25/4) = √25 ÷ √4 = 5 ÷ 2 = 2.5.

Power property: (√a)² = a. Taking the square root and squaring are inverse operations.

Negative numbers: Even roots (2nd, 4th, 6th) of negative numbers are not real. Odd roots (3rd, 5th, 7th) of negative numbers are negative. For example, ∛(-8) = -2.

Common Square Roots to Remember

Perfect squares are numbers whose square roots are whole numbers: √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.

Knowing these perfect squares helps with mental math and simplifying radical expressions. Most other square roots result in irrational numbers (non-repeating, non-terminating decimals) like √2 ≈ 1.414 or √3 ≈ 1.732.

Applications of Square Roots

Square roots appear throughout mathematics and science. In geometry, the Pythagorean theorem (a² + b² = c²) requires square roots to find side lengths of right triangles. In statistics, standard deviation calculations use square roots. In physics, many formulas involving area, energy, and wave mechanics incorporate square roots. Financial calculations for compound interest and investment growth also utilize roots for determining time periods or rates.

For the inverse operation, use the Exponent Calculator to compute powers. Square roots are essential for the Right Triangle Calculator and Distance Calculator.