Squares and Square Roots

Squares & Square Roots

Hello Readers,

Today’s Topic is Squares & Square Roots. In this topic, we will discuss how to find out Squares and Square Roots (Both Perfect and Imperfect).

Before discussing this topic – Squares & Square Roots, it is advisable that you must learn squares up to 30, as it will help you to make your calculations stronger. So, here is the table showing Squares up to 30.

 

Squares & Square Roots - Square of numbers

The only method to learn these squares is Revision. Make your habit to revise these squares once in a day (at least) till you memorize them all.

Now, we will discuss how to solve the square of the numbers up to 100. We have discussed this in two parts.

 

PERFECT SQUARES

1 Part – Squares of numbers from 30 to 80

2 Part – Squares of number from 80 to 130

 

  • 1 Part – Numbers from 30 to 80

Let’s discuss how to find out squares of numbers from 30 to 80.

Here are the steps:

  • First, take Base as 50 (as these numbers, i.e. 30-80 are close to 50)
  • Then apply;

Squares & Square Roots 2

Got confuse after seeing the steps. Don’t worry, we will discuss these steps with the help of examples.

So, here is the example with  explanation.

 

Example

58^2 =?

Traditional Method =  58×58

But we all know it takes, or we can say it wastes too much time in the examination hall.

So here is the Quick Method:

As we have discussed, first we will take Base as 50.

 

Step 1

58 = 50 + 8                                                           Sq 3

 

Step 2

Sq 4         Sq 5

 

Apply this in the above example:

Sq 6         Sq 7

 

Isn’t this is so simple and quick.

Let’s see all these steps in brief as we need not to write all these steps. The above are just an explanation and that is why it is looking a little bit lengthy.

 

Sq 8          Sq 9

 

  • Part 2 – Numbers from 80 to 130

Now, let’s discuss the next part. In this we will learn how to find out squares from 80 to 130.

Here are the Steps:

  • First take Base as 100 ( as these numbers i.e. 80-130 are close to 100)
  • Then apply;

Sq 10

The Steps are similar to the above part. We just have to change the Base=100.

Let’s discuss these Steps with the example and its explanation.

 

Example:

86^2 =?

Traditional Method =  86×86

As said above,  it takes, or we can say it wastes too much time in the examination hall.

So here is the Quick Method:

The Steps are similar to the above part. We just have to change the Base=100.

Let’s discuss these Steps with the example and its explanation.

 

Example:

86^2 =?

Traditional Method =  86×86

As said above,  it takes, or we can say it wastes too much time in the examination hall.

So here is the Quick Method:

 

 Step 1                          

 

86 = 100 – 14       Sq 11

 

Step 2

In the second step, we will apply the formula;

 

Sq 12       Sq 13

 

Apply this in the above example:

 

Sq 14       Sq 15

 

So, after these Quick Methods of finding out squares. Let’s learn how to find out SQUARE ROOTS OF NUMBERS having perfect and imperfect squares.

 

SQUARES ROOTS

Square Roots(Perfect or Imperfect) are mostly asked in the questions of the Simplifications or Approximations

So, Let’s start discussing :

 

  • Perfect Square Roots

The trick to find out Perfect Square Roots is basically based on Unit Places of the Squares of the numbers.

Lets observe Unit Places of the Squares of the Numbers (Perfect Squares)

 

Sq 16

 

As you can see that in the above picture, there is different colored ticks in front of numbers. Observe each color – If the numbers is ending with 1, than its squares is ending with 1 or 9.Similarly of the number is ending with 4 than its square is ending with 2 or 8 and so on.

If the number is ending with 5 or 0, than its square is ending with 5 or 0 only, respectively.

Below is the table depicting the Unit Places Observation shown in the above picture.

 

Squares & Square Roots Imperfect Sq 17

Now we will discuss the whole method with the help of Example along with the Step by Step Explanation.

 

Example:

_/¯4489       =?

Step 1

Sq 18      Sq 19

 

Step 2

 

Sq 20     Sq 23

 

Now we have two options with us; 63 and 67.

We have two methods to get the answer from the two options which are discussed below:

Step 3

Method 1 

 

Sq 22                   Sq 23

 

Note – Trick to find Squares of Numbers ending with 5

Sq 24            Sq 25

 

Method 2

 

Sq 26      Sq 27

 

 

IMPERFECT SQUARE ROOTS

In this section we will discuss about Imperfect Square Roots. The trick to find out Imperfect Square Roots is different. So, here is the trick to get Imperfect Squares.

We will discuss this with the help of example:

Example:

_/¯7200 =?

Step 1

Sq 28     Sq 29

 

Step 2

Sq 30     Sq 31

 

Step 3

Sq 32     Sq 33

 

The exact square root of 7200 is 84.85 and we can see, it is very close to the above answer.

Before finishing this topic, it is necessary to know that above discussed tricks have some limitations:

  • It is difficult to know that which method is to be used for the asked questions as there is different methods for finding Perfect Squares and Imperfect Squares. We have observed from the previous exam pattern that generally or can say most probably Perfect Squares are asked in Simplification problems whereas Imperfect Squares are asked in Approximations.
  • We can also guess by checking some numbers as Perfect Squares have even number of Zeroes.

With this, we will finish our today’s topic. We hope you have enjoyed learning.

 

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Happy Learning !

 

Check out other topics also – 

Simplification – General Rules for Multiplication

Multiplication (Special Cases)

 

Study Notes for all Subjects – 

Bank Exams Study Notes

SSC Exams Study Notes