Online Calculator to Find Square Root

Finding square roots is a fundamental mathematical operation with applications in geometry, algebra, and many scientific fields. This guide explains how to calculate square roots, when they're needed, and how to interpret the results.

What is a square root?

The 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 denoted with the radical symbol (√) before the number.

Square roots can be either positive or negative, but by convention we typically refer to the principal (non-negative) square root unless specified otherwise. For example, √9 = 3, not ±3.

Square roots of negative numbers are complex numbers, which involve the imaginary unit i (where i² = -1). This calculator focuses on real square roots (non-negative results).

How to calculate square roots

There are several methods to find square roots:

Using a calculator: The quickest method for most practical purposes.
Prime factorization: Break the number into prime factors and pair them to find the square root.
Long division method: A manual calculation technique for numbers without perfect square factors.
Estimation: Approximate the square root by finding numbers that, when squared, are close to the target.

For most practical purposes, using a calculator is the most efficient method, especially when dealing with non-perfect squares or large numbers.

Square root formula

Mathematical formula

The square root of a number x can be expressed as:

√x = y, where y × y = x

The square root function is the inverse of squaring a number. It's a strictly increasing function for non-negative numbers, meaning larger inputs produce larger outputs.

Practical examples

Here are some common square root calculations:

√16 = 4 (since 4 × 4 = 16)
√25 = 5 (since 5 × 5 = 25)
√36 = 6 (since 6 × 6 = 36)
√100 = 10 (since 10 × 10 = 100)
√121 = 11 (since 11 × 11 = 121)

For non-perfect squares, the square root is an irrational number. For example:

√2 ≈ 1.414213562
√3 ≈ 1.732050808
√5 ≈ 2.236067977

Frequently asked questions

What is the difference between a square root and a square?

A square is the result of multiplying a number by itself (e.g., 5² = 25). A square root is the inverse operation - finding a number that, when multiplied by itself, gives the original number (√25 = 5).

Can I find the square root of a negative number?

In real numbers, no. The square root of a negative number is not a real number. However, in complex numbers, negative square roots exist using the imaginary unit i (e.g., √-1 = i).

How do I calculate the square root of a fraction?

The square root of a fraction is the fraction of the square roots. For example, √(a/b) = √a / √b.