# How to Find Squares and Square Roots: Step-by-Step Tutorial

- Author: Noreen Niazi
- Last Updated on: August 22, 2023

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**Want to learn how to find squares and square roots? This step-by-step tutorial will guide you through the process and make it easy to understand.**

Squares and square roots are crucial mathematical ideas with numerous practical applications. When a number is multiplied by itself, it produces a square. For instance, 3 x 3 equals 9, therefore 3 squared equals 9. The opposing action is called a square root, and it is the number that, when multiplied by itself, yields the original integer. For instance, 3 is the square root of 9, as 3 x 3 equals 9.

Using various techniques and tools, you will discover how to find any positive number squares and square roots in this course. Additionally, you’ll discover various square and square root properties and rules and how to apply them to issues.

After finishing this tutorial, you will be able to:

- Determine the squares and square roots of whole integers and decimals.
- Find squares and square roots using a calculator or computer.
- When writing squares and square roots, use the exponent notation.
- Prime factorization can be used to simplify square roots.
- Using approximation, calculate square roots.
- Identify perfect squares and irrational numbers.

## What is a square?

A square is a number that is multiplied by itself. **For example**, 3 squared (written as 32) is equal to 9 because 3 x 3 = 9. It is called a square because you can visualize it as a square.

## What is a square root?

A square root is the opposite of a square. When multiplied by itself, it is a number that equals the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

## How to find the square of a number?

To find the square of a number, simply multiply the number by itself.

For example, to find the square of 5, you would do 5 x 5 = 25.

## What is the square root symbol?

The symbol denotes the square root symbol or square root sign ‘√’. It is a mathematical symbol used to represent square roots in mathematics. The square root symbol (√) is also called Radical. For example, we write the square root of 4 as √(4). It is read as root 4 or the square root of 4.

## How to find the square root of a number?

To find the square root of a number, you can use a calculator or do it manually. There are several methods to find the square root of a number. Let’s explore them all one by one.

Read more about decimals and fractions.

Converting Mixed Numbers to Improper Fractions: A Step-by-Step Guide

## Square Root by Prime Factorization:

Prime factorization is the easiest method to find the square root of a number. For this, the number must be a perfect square[1].**Step I**: Resolve the given number into prime factors.**Step II:** Make pairs of similar factors.**Step III: **Take the product of prime factors, choosing one factor out of every pair.

## Solved Examples

*Find the square root of 484 by the prime factorization method*

**Solution:**Resolving 484 as the product of primes, we get

484 = 2 × 2 × 11 × 11

√484 = √(2 × 2 × 11 × 11)

= 2 × 11**Therefore, √484 = 22**

*Find the square root of 324.***Solution:**

The square root of 324 by prime factorization, we get

324 = 2 × 2 × 3 × 3 × 3 × 3

√324 = √(2 × 2 × 3 × 3 × 3 × 3)

= 2 × 3 × 3**Therefore, √324 = 18**

*3. Find out the square root of 1764.***Solution:**

The square root of 1764 by prime factorization, we get

Source: Cue math

1764 = 2 x 2 x 3 x 3 x 7 x 7.

√1764 = √(2 x 2 x 3 x 3 x 7 x 7)

= 2 x 3 x 7

## Practice Problems

Find the square root for the given numbers by prime factorization.

- √625
- √144
- √64

## Square Root by Repeated Subtraction Method:

The sum of the first *n* odd natural numbers is *n^**2*. We will use this fact to find the square root of a number by repeated subtraction. Let us take an example to learn this method. Say, you are required to find the square root of 121, that is, √121. The steps are

- 121 – 1 = 120
- 120 – 3 = 117
- 117 – 5 = 112
- 112 – 7 = 105
- 105 – 9 = 96
- 96 – 11 = 85
- 85 – 13 = 72
- 72 – 15 = 57
- 57 – 17 = 40
- 40 – 19 = 21
- 21 – 21 = 0

Thus, we have subtracted consecutive odd numbers from 121 starting from 1. 0 is obtained in the 11th step. So we have √121 = 11.

## Practice Problems

Find the square root for the given numbers using repeated subtraction:

- √625
- √144
- √64

## Square Root by Long Division Method:

**Example**

**Let us find the square root of 104976 step by step using the long division method.**

**Step 1:**

Separate the digits by taking commas from right to left once in two digits.

10,49,76

When we do so, we get 10 before the first comma.

**Step 2:**

Now we have to multiply a number by itself such that the product ≤ 10

(The product must be greatest and also less than 10)

The above condition will be met by ‘3’.

Because 3 ⋅ 3 = 9 ≤ 10.

Now this situation is explained using long division.

Source Math-only math

In the above picture, 9 is subtracted from 10, and we get the remainder 1.

**Step 3:**

As shown in the picture below, we must bring down 49 and the quotient 3 to be multiplied by 2.

**Step 4**

## Square Root by Long Division Method: Practice Probelms

**Find the square root of the following numbers using the long division method.**

1) 12321

2) 16384

3) 65536

4) 47472

## Square Root by Estimation Method:

One method is to guess and check.

- Start with a number and square it.
- Try a smaller number if the result is greater than the original number.
- Try a larger number if the result is less than the original one.
- Keep doing this until you find the exact square root.

## How to find the square of a negative number?

Just like positive numbers, we can find squares of a negative number by similar methods. Multiply the negative number by itself. We get a positive number.

**For example**: Multiply -5 with -5 we get 25.

## What are perfect squares?

Perfect squares are numbers that have whole number square roots[2]. For example, **4, 9, 16, and 25** are perfect squares because their square roots are **2, 3, 4, and 5,** respectively.

## How are squares and square roots used in real life?

Squares and square roots are used in many areas of life, including math, science, and engineering. They are used to calculate areas of squares and rectangles, as well as volumes of cubes and other three-dimensional shapes. They are also used in physics to calculate velocity and acceleration and in finance to calculate interest rates and investment returns.

## Conclusions and Main Points About Squares and Square roots

Finding squares and square roots is a crucial mathematical skill that can assist us in resolving a wide range of issues, including area, distance, the Pythagorean theorem, quadratic equations, and more.

We can use various techniques to find squares and square roots, including repeated addition, factorization, long division, estimate, and a calculator. The method we choose is determined by the kind and amount of the numbers we are dealing with and the required accuracy level. Similar ideas that involve multiplying an integer by itself include squares and square roots. Taking the square root of a number yields the side length of a square with that area while squaring a number yields the area of a square with that side length.

We must comprehend their meanings and connections in order to identify squares and square roots. Finding a number’s square root entails discovering the number that was multiplied by itself, whereas squaring a number entails multiplying it by itself. There are several uses for squares and square roots in the domains of geometry, mathematics, physics, and other subjects. To find squares and square roots, we can employ various methods, including factoring, prime factorization, the square root algorithm, approximation using decimals or fractions, and a calculator. The best course of action depends on the number’s size and the precision level we require.

## FAQs

A number is multiplied by itself to produce the square of the number. For instance, 3 multiplied by 3 is 9, the square of 3. This is spelled out as 32=3*3=9. “Squared” or “to the power of 2” are denoted by the symbol “2”.1

There are various ways to calculate a number’s square root. Some prominent techniques include:

- Applying the equation

$$y.y=y^2=a,$$

- where ‘a’ is the square of the value ‘y’1.

Finding the square root through division - Utilizing techniques including the prime factorization method, the method of repeated subtraction, and the long division method.

- No, square root and square are not equal. In mathematics, they are the opposing operations. While finding a number that, when multiplied by itself, returns the original number yields the square of a number, finding a number that, when multiplied by itself, returns the original number yields the square root of a number.
- For instance, since 42=4*4=16, the square of 4 is 16. Since 16 equals 4 and $$4^2$$ equals 16, the square root of 16 is 4.

The distinction between a perfect square and a square root is that the former can be expressed as the product of two equal rational numbers. In contrast, the latter can be expressed as the product of two equal numbers that result in the original number.3

**For instance**, 25 is a perfect square since it can be expressed as 55=25. Because 55=25 and (5)(5)=25, respectively, the square roots 25 are 5 and -5.

Not every number forms a perfect square. It is not a perfect square because it cannot be expressed as the product of two equal rational integers, for instance, 8. Because 8’s square root is an irrational integer, it cannot be expressed as a fraction. About 2.828 is the equivalent.4