Squares and Square Roots

• Introduction
• What is a Square?
• How Do You Find the Square of a Number?
• Perfect Square Table
• What is Square Root?
• How to Find Square Root?
• Repeated Subtraction Method
• Prime Factorization Method
• Estimation Method
• Long Division Method
• Square Root Table
• Practice Problems
• Frequently Asked Questions

Introduction

In Maths, squares and square roots play an important role in various mathematical operations. A square is obtained by multiplying a number by itself, expressed as "a squared" or \(a^2\), representing the area of a square with side length '`a`' On the other hand, square roots involve finding the number that, when multiplied by itself, equals a given value. These concepts are fundamentally used in algebra, geometry, and numerous real-world applications.

In addition to square roots, there are some different types of roots:

`1`. Square Root: For example  \( \sqrt{25} = 5 \)

`2`. Cube Root: For example \( \sqrt[3]{27} = 3 \)

`3`. Fourth Root: For example \( \sqrt[4]{16} = 2 \)

`4`. `n`th Root: For example \( \sqrt[6]{64} = 2 \) (for \( n = 6 \))

Our focus would be on the square root. 

What is a Square?

In exponents, a square refers to raising a number to the power of `2`. It is denoted by the superscript "`2`" and signifies multiplying the number by itself. For instance, if '`a`' is a number, then "a squared" is written as \(a^2\) and is equal to \(a \times a\). This operation essentially represents the area of a square with side length '`a`' in geometry. Squaring is a basic arithmetic operation that finds applications in various mathematical and scientific calculations.

How Do You Find the Square of a Number?

Finding the square of a number involves multiplying the number by itself. 

Example: Find the square of `7`.

Solution:

`1`. Write down the number: \(7\).

`2`. Multiply the number by itself: \(7 \times 7\).

`3`. Calculate the result: \(49\).

So, the square of `7` is \(49\).

In general, if you have a number '`a`', the square is represented as \(a^2\) and is calculated by multiplying '`a`' by itself: \(a \times a\). This fundamental operation forms the basis for understanding more complex mathematical concepts.

Perfect Square Table

Below, you'll find a listing of perfect squares of numbers ranging from `1` to `30`.

What is Square Root?

The square root operation is the opposite of squaring a number. Squaring involves multiplying a number by itself, while finding a square root involves finding a number that, when multiplied by itself, equals the original number. For example, if we take the square root of `16`, it is `4` because `4` times `4` equals `16`.

The square root is a special symbol `(√)` that looks like a tick, known as the radical. In the world of mathematics, this symbol always adds a sense of importance to the calculations. The number inside the radical symbol is called the radicand. 

Example: Find the value of `sqrt36`.

Solution: 

Here `36` is known as the radicand. When we say "square root of `36`," we're essentially searching for a number that, when multiplied by itself, equals the radicand, `36`. 

For this example, the square root of `36` is `6` because `6` times `6` equals `36`. So, we express it as "the square root of `36` equals `6`." 

Hence  `sqrt36 = 6`.

How to Find Square Root?

The key is to finding the square root of a number is to determine which number, when multiplied by itself, results in the original number. This process is particularly simple for numbers that are perfect squares—those that can be expressed as the product of an integer multiplied by itself.

Perfect squares are special numbers making them easily recognizable. There are primarily four methods to find square roots:

`1`. Repeated Subtraction Method

`2`. Prime Factorization Method

`3`. Estimation Method

`4`. Long Division Method

Each way of finding square roots has its own advantages. The first three methods are great for easy cases like perfect squares. On the other hand, the long division method is the most robust one that can handle any type of number, whether it's a perfect square or not. So, no matter if you're dealing with simple perfect squares or more complex numbers, these methods give you the tools to become a square root expert.

Repeated Subtraction Method

Discovering square roots becomes easy with the Consecutive Odd Numbers Method. This method is designed specifically for figuring out the square root of perfect square numbers. Let's go through the steps using the example of finding the square root of `25`:

Step `1`. Commence Subtraction: Initiate the method by subtracting consecutive odd numbers from the given number until you hit `0`.

Step `2`. Tally the Subtractions: Keep a count of how many times you carry out the subtraction. This count signifies the square root of the given number.

Example: Find the square root of `25` using the consecutive ddd numbers method.

Solution:

\(25 - 1 = 24\)

\(24 - 3 = 21\)

\(21 - 5 = 16\)

\(16 - 7 = 9\)

\(9 - 9 = 0\)

By noting the number of subtractions (`5` times in this case), you determine that the square root of `25` is `5`, as \(5 \times 5 = 25\).

This method proves handy when dealing with not so big numbers like `25`. However, please note that consecutive odd numbers method only works for perfect square numbers.

Prime Factorization Method

The Prime Factorization Method is a handy tool that makes finding the square root of a number, especially if it's a perfect square, much easier. Let's go through the steps with a simple example of finding the square root of `36`.

Example: Find the square root of 36 using the prime factorization method.

Solution:

Step `1`. Identify Prime Factors: Express the given number `36`, as a product of its prime factors. In this case, \(2 \times 2 \times 3 \times 3\).

Step `2`. Pair Up Prime Factors: This gives us \((2 \times 2) \times (3 \times 3)\).

Step `3`. Extract the Square Root: Take one factor from each pair. We will then have \(2 \times 3\).

Step `4`. Multiply the Extracted Factors: \(2 \times 3 = 6\)

So, the square root of `36` is `6`. This can be verified because \(6 \times 6 = 36\).

The prime factorization method simplifies the process by breaking down the number into its prime components, making it easier to identify and extract the square root.

Estimation Method

Estimation and approximation involve making educated guesses in math to simplify calculations. Let's apply this method to find the square root of `21`.

Example: Find the square root of `21` using the estimation method.

Solution:

Identifying the perfect square numbers nearest to `21`, we have `16` and `25`. The square root of `16` is `4`, and the square root of `25` is `5`. So, the square root of `21` lies between `4` and `5`.

To narrow it down, we check between `4.5` and `5`. Since `4.5` squared is `20.25` and `5` squared is `25`, we see that the square root of `21` is closer to `4.5`. Now, for a more precise answer, we look between `4.5` and `4.6`. The squares of `4.5` and `4.6` are `20.25` and `21.16`, respectively. The square root of `21` is between `4.5` and `4.6`.

Refining further, we check between `4.55` and `4.6`. After calculations, we find that the square root of `21` is approximately `4.58`. This process can be quite time-consuming, however, it caters to all numbers irrespective of them being perfect or non-perfect square numbers.

Long Division Method

The long division method is a systematic approach for breaking down complex division problems, making it particularly useful for finding square roots. As well, it works to find the square root of to all numbers irrespective of them being perfect or non-perfect square numbers.Let's explore this process using the example of finding the square root of 576. 

Example: Find the square root of `576` using the long division method.

Solution:

Step `1`: Pairing Digits: Place a bar over each pair of digits from the right, starting at the units' place. For `576`, we have two pairs: `5` and `76`.

Step `2`: Initial Division: Divide the leftmost pair `(5)` by the largest number whose square is less than or equal to `5`. In this case, the largest such number is `2`, resulting in a quotient of `2`.

Step `3`: Bringing Down and Expanding: Bring down the number under the next bar `(76)` to the right of the remainder. Add the last digit of the quotient `(2)` to the divisor `(2)`, forming a new divisor.

Step `4`: Refining the Quotient: Select a suitable digit to add to the divisor, making a new divisor for the carried-down dividend. The new digit in the quotient will be the same as the one selected in the divisor, maintaining the condition of being less than or equal to the dividend.

Step `5`: Decimal Point and Zeros: Continue the process with a decimal point and add zeros in pairs to the remainder.

Step `6`: Obtaining the Square Root: The quotient obtained at this stage is the square root of the given number. For `576`, the square root is exactly `24`. This method allows us to break down the complex process of finding square roots into manageable steps. In each step we are dealing with smaller numbers compared to the original dividend.

Square Root Table

The square root table is a useful reference containing numbers and their corresponding square roots. This table is particularly handy for determining both the square roots of perfect square numbers and calculating the squares themselves. Here, you'll find a list of square roots for perfect square numbers and a selection of non-perfect square numbers between `1` and `10`.

Practice Problems

Q`1`. What is the square of `8`?

1. `16` 
2. `32` 
3. `64`  
4. `128`  

Answer: c

Q`2`: Find the square of `5`.

1. `10`  
2. `15`  
3. `20`  
4. `25` 

Answer: d

Q`3`. What is the square root of `81`?

1. `7` 
2. `8`  
3. `9`  
4. `10` 

Answer: c

Q`4`. Determine the square root of `25`.

1. `3`  
2. `4`  
3. `5`  
4. `6`  

Answer: c

Q`5`. Find the square root of `144`.

1. `10`  
2. `12`  
3. `14`  
4. `16`  

Answer: b

Frequently Asked Questions

Q`1`. What is a square in mathematics?

Answer: A square in mathematics refers to the result of multiplying a number by itself. For instance, the square of `4` is `16` because `4` multiplied by `4` equals `16`.

Q`2`. How do I calculate the square of a number?

Answer: To calculate the square of a number '`a`', simply multiply '`a`' by itself, denoted as \(a^2\).

Q`3`. What is a perfect square?

Answer: A perfect square is a number that can be expressed as the product of an integer multiplied by itself, resulting in a whole number. Examples include `9 (3×3)` and `25 (5×5)`.

Q`4`. How do I find the square root of a number?

Answer: To find the square root of a number '`b`', look for a number that, when multiplied by itself, equals '`b`'. This is denoted as \(\sqrt{b}\).

Q`5`. Can negative numbers have square roots?

Answer: Yes, negative numbers have square roots. However, we do not use real numbers to write the square root of negative numbers. Instead we use complex numbers to write square roots of negative numbers.