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Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The square root is essential in various fields such as construction, engineering, design, finance, and navigation. In this topic, we are learning about the square root.

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Professor Greenline from BrightChamps

What is a Square Root in Math?

In mathematics, a square root is a fundamental concept. A number's square root is a value that yields the original number when multiplied by itself. Look at this formula (y2= x), now we can say that the square root of x is a number we get by multiplying y by itself (y2= x). Just the same, we can figure out the square root of any number. Any positive real number has one positive and one negative square root. For instance, the square root of 9 is ±3, where 3 x 3 = 9 also (-3) x (-3) = 9. 

Professor Greenline from BrightChamps

History of Square Roots

The history of square roots dates back thousands of years. It shapes young minds in  understanding of the concept as we know it today. Ancient civilizations, like the Babylonians and Egyptians, were pioneers in developing the square roots of mathematics. They even developed efficient methods to approximate the value.

 


By around 500 BCE, the Pythagoreans employed square roots for their works on geometry. During the medieval and modern eras, with the advent of algebra, square roots became more widespread. In 1450 AD, Regiomontanus invented a symbol, written as ‘R’ to denote square roots. Years later, in 1525, Christoph Rudolf introduced the now-familiar symbol √ for square root. Today, square roots are used in countless real-life applications such as calculating dimensions, forces, distances, and so on.  
 

Professor Greenline from BrightChamps

Properties of Square Roots

The square root has several properties that make it both unique and easy to understand. Let us uncover these properties:

 

  • A perfect square number has a perfect square root. 16 is a perfect square, since it is equal to 4 × 4. Additionally, ±4 is the square root of 16. 

     
  • If the last digit of a number is an even number of zeros, it can have a square root. Take a look at this, the square root of 100 is √100 = 10 (10 × 10 = 100).

     
  • The values of two square roots can be multiplied. For example, √3 × √2 = √6. 

     
  • If we multiply two same square roots, we get a radical number, such as √3 × √3 = 3. 

     
  • If we multiply a negative square root by itself, it does not give a negative result. Instead, it yields a positive number. For instance, (-3) × (-3) = 9. 

     
  • Numbers ending with 2, 3, 7, or 8 do not have a perfect square root. There is no whole number whose square equals 32, 53, 77, or 98. 

     
  • Numbers ending with 1, 4, 5, 6, or 9 in the unit place have the perfect square roots. For example, ±5 is the square root of 25.
     
Professor Greenline from BrightChamps

Classification of Square Roots

The square root of a number can be classified into two types: perfect squares and non-perfect squares. Let us now understand more about the classifications. 
 

Professor Greenline from BrightChamps

Perfect Squares

A perfect square is said to be an integer that is the square of another integer. In simple terms, it is a number resulting from squaring another number. As an example, 5 × 5 results 25. Here, 25 is a perfect square and it is the product of 5. Also, 5 is an integer. 


Some examples of perfect squares are: 


1 = 1 × 1 = 12
4 = 2 × 2 = 22
9 = 3 × 3 = 32

16 = 4 × 4 = 42

25 = 5 × 5 = 52

36 = 6 × 6 = 62

 

These examples clearly illustrate the concept of a perfect square. A perfect square is the result that is obtained by multiplying an integer by itself.
 

Professor Greenline from BrightChamps

Non-Perfect Squares

A non-perfect square root is not an integer. It will be an irrational number instead. It cannot be expressed as a simple fraction. They are numbers that do not result from multiplying an integer by itself.  Its decimals never stop.

 

For example, the square root of 2 is:


√2 ≈ 1.414


The square root of 2, which is 1.414 is an irrational number. Non-perfect squares have square roots whose values are not whole numbers. A few examples of non-perfect squares are:


√2 ≈ 1.414
√3 ≈   1.732
√5 ≈ 2.236
√8 ≈ 2.828


In advanced mathematics, non-perfect squares are widely used, where approximate values are required. 
 

Professor Greenline from BrightChamps

Importance of Understanding Square Roots for Students

Let us dive into the world of square roots and understand how it works. It has countless applications in everyday life. Square roots help students to solve academic and real-world problems. Some key benefits of mastering square roots by students are:

 


Root for advanced math


In advanced mathematics, square roots are a fundamental element. In algebra, geometry, and calculus they play a significant role in solving quadratic equations and comprehending logarithms and exponents. 

 


Real-world uses 


Students can apply square roots in different areas such as architecture, construction, engineering, and physics. It is important in calculating distances, forces, and intensity of energies. Also, it plays a crucial role in financial contexts, such as in determining compound interests and analyzing the risks in financial properties. 

 


Encourage math confidence 


Understanding square roots helps students enhance their critical thinking and analytical skills. It boosts their confidence in solving challenging math problems and strengthens their confidence in the subject. This foundation enables them to focus on future studies that heavily rely on square roots and mathematics.
 

Professor Greenline from BrightChamps

Methods to Calculate Square Roots

There are different methods we use to calculate square roots of a given number. Some of the methods we use are given below:

Professor Greenline from BrightChamps

Prime Factorization Method

While using the prime factorization method for square root calculation, a number is represented as the product of its prime factors. Remember that this method is only applicable to perfect square numbers. To find the square root of a perfect square, the prime factorization is the most commonly used method. We need to follow the steps mentioned below to calculate the square root of a number:

 


Step 1: Break down a number into its prime factors. Start with the smallest prime number, which is 2. 

 


Step 2: Group pairs of factors where both factors in each pair are the same. 

 


Step 3: Take one factor from each pair. 

 


Step 4: Multiply the factors. The square root of a given number is the product of their factors.


For instance, take a look at this example of the square root of 144:


144 = 2 × 2 × 2 × 2 × 3 × 3
(2 × 2) × (2 × 2)  × (3 × 3) 
(2 × 2 × 3)2 
144 = 12 × 2

√144 = 12


The square root of 144 is ±12. 
 

Professor Greenline from BrightChamps

Long Division Method

To find the square root of an imperfect number we can apply the long division method. In this method large numbers will be broken down into small parts.  The several steps of this method are:

 


Step 1: Break a number into pairs of two digits from right to left. 

 


Step 2: Find the greatest number whose square is smaller or equal to the first digit or pair.

 


Step 3: Now we can subtract the square of that number from the pair. After that, we can drop the next pair of numbers. 

 


Step 4: Double the number you found in step 2, and you get the new divisor.

 

 
Step 5: When we reach the required level, we can stop the steps. 

 


For example, we can find the square root of 20:

 


Step 1: We have one pair of digits for 20. 

 


Step 2: 4 is the largest number whose square is smaller than or equal to 20, because (4 × 4 = 16). The first digit of the square root of 20 is 4. 

 


Step 3: Subtract 16 from 20. 
20 - 16 = 4
Now, we have to drag down the next pair. But here, there is no other pair. So add a zero and make it 400. 

 


Step 4: Double the digit 4 and we get 8 as the result. 

 


Step 5: Next, we find the largest number, whose square is less than or equal to 400, when we multiply by the new number and add to 8. Here, 5 is the next digit. 
8 × 5 = 40
40 × 5 = 200

 


Step 6: Subtract 200 from 400 which gives 200. 
Now, we know the approximate square root of 20 is ±4.5. 
 

Professor Greenline from BrightChamps

Using a Calculator

This is a simple and interesting way to calculate the square roots of any given number. Finding the symbol of square root √ in the calculator is the foremost thing you should remember. Just enter your number then press the square root symbol √. The calculator will display the square root. 

Professor Greenline from BrightChamps

Tips and Tricks to Master Square Roots

Some tips and tricks will help students to understand more about the square roots. Also, it makes the calculations easier. After applying these tips and tricks, kids can solve difficult mathematical problems and equations. 

 


Learn the basics:

 

Primarily, students should study the fundamentals of square roots. It is a value that, when we multiply by itself, we get the original number. 

 


Remember perfect squares:

 

Another useful trick is to by heart the perfect squares of numbers from 1 to 15. For example, we know that, 22 = 4 and 32 = 9. Memorizing these will help students quickly identify perfect squares during calculation.

 


Identify the nearest perfect squares:

 

For non-perfect squares, estimate the nearest perfect squares. For instance, to identify the √20, find the closest perfect squares, they are √16(4) and √25(5). 

 


Understand the prime factorization:

 

To simplify the square roots of any given number, break the number into its prime factors. It will help the students to solve problems more easily and faster. 

 


Apply division method:

 

If the given number is large, kids can use the division method to find the square root. By breaking a large number into smaller parts, the calculations become much easier.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Square Roots

While finding the square roots of any number, the calculations might seem tricky. Students may also make several common errors during the solving process. Here are some common mistakes and their solutions. Avoiding these errors will help students make accurate calculations.
 

Mistake 1

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Confusing square roots with squares

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Always remember, that the square root of a number is equal to a number we get due to squaring the value. Also, the square is the result we get when we multiply a number by itself. For a better understanding, take a look at this:


The square root of 16 is ±4 because 4 × 4 = 16


The square of 4 is 16 because 42 = 16. 
 

Max from BrightChamps Saying "Hey"

Solved Examples on Square Roots

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Bobby is planting flowers in a square garden with an area of 169 square feet. What is the length of one side of the garden?

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The length of one side of the garden is 13 feet

Explanation

To find the length of one side, we need to calculate the square root of 169.


The formula to calculate the area is:


Area = side × side. It is a square-shaped garden, so all four sides are equal. Hence, 


Side = √Area


Side = √169


Now, we have to find the √169.


13 × 13 = 169 


Therefore, Side = 13
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

Find the value of ‘x’ in the equation: √(x+2) = 4

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 x = 2 or x = -6. 
 

Explanation

To find the value of ‘x’, in √(x+2) = 4 


we need to square both sides:


√(x+2) = 4
x+2 = 4
x= 4 - 2


Thus, x = 2 or x = -6. 
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

What is the square root of the sum of 10 and 6?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The square root of the sum of 10 and 6 is √16, which equals 4. 
 

Explanation

First, we have to calculate the sum of 10 and 6:


10 + 6 = 16


Then, find the square root of 16:


√16 = 4

 The square root of the sum of 10 and 6 is √16, which equals 4. 
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

Joyal is building a square path in his garden. If the area of the path is 100 square feet, what is the length of one side of the path?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

To calculate the side length, find the square root of 100.


√100 = 10
 

Explanation

 The square root of 100 is 10. Therefore, the length of one side of the path is 10 feet. 
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

What is the square root of the product of 18 and 2?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The square root of the product of 18 and 2 is √36, which equals 6. 
 

Explanation

First we need to multiply the given numbers:


18 × 2 = 36


Now, find the square root of 36:


√36 = 6


The square root of the product of 18 and 2 is √36, which equals 6. 
 

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQs on Square Roots

1.What is a square root in Math?

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2.Is the square root of 169 a whole number or not?

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3.What are the methods to find the square root of a number?

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4.How can one calculate the square root of a perfect square?

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5.How to determine the square root of a non-perfect square?

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6.What is the square root of zero?

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7.Can square roots be negative?

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8.Can negative numbers have square roots?

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9.What is a perfect square?

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10.What is the significance of square roots?

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11.What is the square root of 225?

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