Square Root Calculator

Use our versatile square root calculator to instantly find the square root of any given number. Whether your input is positive or negative, this root calculator will accurately identify both the principal square root and its opposite (negative) counterpart.

Directions for use

Using our square root calculator is simple and intuitive. Just enter your target number into the input field and click “Calculate.” The tool will instantly return the principal square root alongside the opposite (negative) square root. Additionally, it will clearly indicate whether your input is a perfect square.

Squares and square roots

Squares

A square of a number is the result of multiplying that number by itself. For example, 3 × 3 = 9. This means the square of 3 is 9, or "three squared" equals 9. Mathematically, the square of a number is typically written as x². So, if x = 3, the equation is expressed as 3² = 9. Below are some common examples of squared numbers:

Now, let’s look at negative numbers. What is (-3)²? Since multiplying two negative numbers yields a positive result, (-3)² = (-3) × (-3) = 9. Therefore, (-3)² = 3² = 9.

Perfect squares

A perfect square is the exact square of an integer. For instance, 4, 9, 16, and 25 are all perfect squares. Memorizing the perfect squares of the first few integers is highly beneficial for solving math problems quickly. Here is a helpful reference table:

Consequently, if the square root of a number results in a whole integer, that original number is classified as a perfect square. Our handy calculator will always notify you if your input falls into this category.

Square roots

The square root of a number is a specific value that, when multiplied by itself, produces the original number. For example, the square roots of 9 are 3 and -3, because 3 × 3 = 9 and (-3) × (-3) = 9. Thus, (-3)² = 3² = 9. Similarly, the square roots of 16 are 4 and -4. Every positive real number has exactly two square roots: one positive and one negative.

The positive square root is officially known as the principal square root. When a math problem doesn't specify which root to find, it is universally implied that you should calculate the principal root. For example, if asked, “What is the square root of 36?”, you are only expected to provide one answer. Therefore, you take the principal root, and the answer is 6.

The square root symbol

The square root symbol is called a radical and is represented by the √ sign. To write the square root of 16 mathematically, you would write √16.

According to the strict mathematical definition of a function, for any function y = f(x), there must be a unique, single value of y for every value of x. If we had a function where y equals the square root of x, each x would produce two y values (one positive and one negative). This directly contradicts the core definition of a function! To resolve this, mathematicians assigned the radical symbol (√) exclusively to the principal (positive) root.

This means that while the actual square roots of 16 are 4 and -4, the mathematical expression is strictly √16 = 4. You must keep this in mind when solving equations. Any equation in the format of y² = x will always yield two solutions, properly written as y = √x and y = -√x, or simply y = ±√x.

Square roots of negative numbers

As established earlier, the square of any real number is always positive. A positive number squared is positive, and a negative number squared is also positive (since two negatives cancel out).

But what if a number produces a negative result when squared? Numbers that yield negative results upon squaring are classified as imaginary numbers. The foundational imaginary number is represented by i, which is defined as:

i² = -1

or

i = √(-1)

Let’s apply this to find the square roots of (-4):

√(-4) = √(4 × (-1)) = √4 × √(-1) = 2 × i = 2i

The principal square root of (-4) is 2i. If we also consider the opposite (negative) square root of 4 (-√4 = -2), we get the secondary solution: -2i.

How to find the square root of a number

Calculating the square roots of perfect squares is straightforward. However, finding the square roots of decimals or non-perfect integers manually can be quite complex. While various manual methods exist to calculate exact roots, using an online square root calculator is the fastest, most reliable way to find precise answers instantly.

Real-life application

Let’s look at a practical example. John is looking to rent a studio apartment and finds a listing advertising a space with an area of 20.25 square meters. How can he estimate the length of the walls to better visualize the room's actual size?

Solution

In real estate, property sizes are primarily listed in square meters. While some listings provide the precise wall lengths, many do not. It can be difficult to picture a space solely based on its total square meterage. However, if we imagine the total area as a perfect square room, we can easily estimate the dimensions. To find the length of one side, we simply need to extract the square root of the total area:

√20.25 = 4.5

Since we are calculating the physical, real-world dimensions of an apartment, we only need the positive (principal) square root.

Interestingly, extracting square roots also applies directly to units of measurement! In this example, the total area was measured in square meters (m²). When calculating the wall length, we are technically taking the square root of 20.25 m²:

√(20.25 m²) = √20.25 √(m²) = 4.5 m

Answer

A studio apartment with an area of 20.25 square meters can be effectively visualized as a square room where each wall measures precisely 4.5 meters long.