Squares and squares root class 8 :[Notes,worksheet and their PDF]

Here, you will get “Square and Square Root Class 8 Notes, Worksheets with their PDF.So that you can prepare the full concept in a single place. Let’s Start 

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Definition AND Types of Squares 

           What is the square of any number mean?

When we multiply a number by itself, then the product is called the square of that number.

For example: Suppose I want to find the square of 5, then I’ll multiply 5 to itself i.e 5×5=25 . Hence 25 is the required square of 5.

Here’s the formal definition of square

If x and y are two natural numbers, then y=x² means y is the square of x .

I think, now it is clear. So, let’s see types of square

Types of Squares 

All Squares is divided into two categories on the basis of their nature.

  • Perfect square
  • Non-perfect square

What is a perfect square?

A Square formed by multiplication of the whole numbers is called a perfect square.

For Example, The Square of 5 is 5×5=25 where 5 is a whole number. So, 25 is a perfect square. Right

Important Notes:

(i)  All natural numbers are not perfect squares.(Example :3,5,7,11,13 etc are not perfect squares ) 

(ii) There are only 10 numbers up to 100 which are perfect squares. You can see the numbers given in the below list.

Here is a list of the square from 1 to 30. These are perfect squares 

NUMBER SQUARE(Perfect Squares)
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625
26 576
27 729
28 784
29 841
30 900

What are non-perfect squares? 

To understand the concept of “Non-Perfect Squares”, you must have knowledge of square root which we’ve covered later in the Post. So, If you’ve never read about Square Roots, SKIP to the Square root section, learn it first then, read this section.

Definition of Non-Perfect Squares 

It is an integer whose square root is not a whole number .

For Example, It contains numbers other than perfect squares such as √3,√5,√13, etc.

10 properties of the perfect squares

1. A number ending with 2,3,7 or 8 is never a perfect square. So,22,43,27,38…are not perfect squares because the ending digits are 2,3,7, and 8. But, it has some exceptions as well which you can found in the notes given below

Exception:  It is not necessary that numbers ending with digits 0,1,4,5,6 or 9 will be perfect squares. For example, 184 is not a square number even it is ending with the digit 4. 

2. A number ending with an odd number of zeros is not a perfect square.

For example: 50,9000,1600000 are not perfect squares.

Exception: A number ending with an even number of zeroes need not be a perfect square .e.g 500 is not a perfect square.

3. Square of an odd number is odd.

For example: 7²=49 ,13²=169 .We know 49  and 169 both are odd numbers.

4. Square of an even number is even.

For example: 12²=144 ,22²=484.Here 144 and 484 both are even numbers.

5. For every natural number n,

The difference between the squares of two consecutive numbers is equal to the sum of 
the numbers .

For Example: Let’s take two consecutive numbers 12 and 13. Find the Difference between their squares 

13² – 12²=169 -144 =25 …(i) 

Now add given numbers =12+13=25 ,it is also 25. This is what the given statement is.

6. Pythagorean triplet

What is the Pythagorean triplet?

Pythagorean triplets are three numbers that satisfy the Pythagoras theorem. For example, 3,4 and 5 are Pythagorean triplets

Because It satisfies Pythagoras theorem 

5²=3²+4²

General formula for Pythagorean triplets for every natural number n where n>1
is (2n,n²-1,n²+1)

With the help of given General  formula, you can find as many triplets you want but keep in mind “Value of n should be greater than 1”.The smallest Pythagorean triplets are 3,4 and 5 which you can be calculated by substituting n=2. And we can find more triplets by substituting n=3,4,5,etc .

7. On dividing a perfect square by 3, we get 0 or 1 as the Remainder.

For Example : 

4=3×1+1

9=3×3+0

16=3×5+1

We have 4,9 and 16 as perfect squares .On dividing it by 3 ,we get 0 or 1 as Remainder .You can check it with more perfect squares .

8. If n is a perfect square, then 2n can never be a perfect square.

For example : 16 is a perfect square but 2×16=32 is not a perfect square.

9. Here is an amazing relation between perfect squares and squares of digits of series “1” 

         1²=1                           1=1=1²

        11²=121 ,                    1+2+1=4=2² ⇒ Sum of digits of square =Perfect square 

       111²=12321             1+2+3+2+1=3² ⇒                     "
                ...
                    ...

                    ...
   111111111²=12345678987654321 ,       1+2+3+4....4+3+2+1=81=9²

10. There are 2n numbers ly between n² and (n+1)²

For Example : Let n=3 

So, n+1 = 3+1 =4 

Now Square of given numbers are n² =(3)²=9 and (n+1)²=(4)²=16 

Numbers lying between (3)²and (4)²are 10,11,12,13,14,15.They are total 6 in collection .

The Value of 2n= 2×3=6 which is equal to Total number lying between (3)² and (4)².

Definition of square roots

The square root of a number is nothing but the opposite of the square. Let’s understand it with an example

We know the square of 6 will be =6×6=6²=36

The reverse of this process will be √36=6 where the” √ “sign denotes the square root of 6.

How to find square root using the prime factorization method

Procedure:

Step 1: Write the prime factors of the given number

Step 2: Make the pairs of equal factors

Step 3: To find a square root write one factor corresponding to each pair

Step 4: Multiply the factors obtained in the above step

Let’s take an example to understand it clearly

Example :  Find the square root of 784

Solution: Prime factorization of 784 =2×2×2×2×7×7

prime factorization of 784

Now, make the pair of each pair 784 =(2×2)×(2×2)×(7×7).

Add square root on both sides

√784=√(2×2)×(2×2)×(7×7).

√784=2×2×7

√784=28

You can check these posts on “Square Root Using Factorization Method “.

How to find square root using the ones and tens method

Procedure:

Step 1: Observe the one digit of the perfect square.As discussed above, perfect square numbers ending with 1,4,6 or 9 will have two possible one digits.

Step 2: Strike out the last 2-digits i.e “O” and “T” digits from the right

Step 3: For tens digit, think of a number whose square is less than or equal to this leftover number.

Step 4: Multiply the fraction obtained in step 3

Here is an Example 

Example : Find the square root of 2304

Sol. Here ,the last digit of 2304 is 4 (So, a number ending with digit 4 will have two possible one’s digits i.e 2 or 8 )

After striking the last two-digit, now is time for the remaining  two digits which is 23

Think 23 lies between which number square?

I guess, it is 4² and 5². Between these 23 lies i.e 4²<23<5²

Now 4² is less than 23.So, the tens digit of the required number is 4 what about one’s digit. It may 2 or 8 as mentioned above

Check whether 42²=2304 or 48²=2304

So, the correct answer is 48

You can check this Post on “Square Using Ones and Tens Method”. 

How to find square root using long -division method

I’ve already created Many Posts on ,”How to find Square Root Using Long Division Method ” ? You can check here 👇

Square Root Word problem solution

Problem: 01 The area of a square is 729 sq m. Find the dimension of the square

Sol:

Area of square=729 sq m.

We know, Area of square=(side)²

∴   Side of a square =√Area

=√729

=√(3×3)×(3×3)×(3×3) m

= 3×3×3=27m

∴ The side of the square measures is 27 m.

Problem:02 Find a number whose one-third multiplied by its one-sixth becomes 162.

Solution:

Let the number be x

According to the question

\frac{x}{3}×\frac{x}{6}=162

\frac{x²}{18}=162

x²=162×18

x²=2916

x=√2916

x=54

The required number is 54

If you want to see more word problem on square root with solution :Check this 

Square and square root class 8 notes PDF

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Square and the square root class 8 :[ worksheet]

Try this square and the square root class 8 worksheet and check your concept which you’ve read above.

(I)  MCQ and word problem

Q.1 Write the unit’s digits of the square of the following numbers

(i) 93                    (ii) 72            (iii) 546                   (iv) 729

Q.2 Find the square of the following :

(i) 47                    (ii) 1.23

Q.3 How many numbers lie between (i) 7² and 8²

Q.4 Using the prime factorization method, examine which of the following numbers are perfect squares:

(a) 632                           (ii) 1458                 (iii) 7225

Q.5 Find the square root of the following numbers using the ones and tens method.

(a) 1521                     (b) 1764

Q.6 Find the dimension of a square whose area is 2025 sq.m

Q.7 Without actually finding the square root of the number, determine the number of digits in the square root of :

(a) 2025                     (b) 271441

Q.8 Find the least number which when added to 4529 makes it a perfect square.

Q,9 Find the smallest 6-digits number, which is a perfect square.

Q.10 Find the greatest 6-digit number, which is a perfect square.

Q.11 Find the √37.22 correct up to two places of decimal.

Q.12  Find the cost of putting a wire along the boundary of a square field of area 1296 m² if the wire costs Rs 25 per m .

Q.13 Find the cost of putting a wire along the boundary of a square field of area 625 hectares if wire costs rs 15 perm.(HINT: 1 hectare=10000m²)

(II) Fill in the blanks

(i) There are _ non-perfect squares between 50² and 51 ²

(ii) Square of 41 is _ (odd/even)

(iii) √529=_

(III) True and false

(i) There are,26 non-perfect squares between 13² and 12²

(ii) The unit’s digit in the square root of 729 is 3

(iii) Squares of 1600 will end with two zeros

(iv) The unit’s digit in the square of 78 is 4

(IV) Give reasons for the given statement

(i) 441000 is not a perfect square.

(ii) 1728 is not a square

(iii) We cannot find the square root of a negative number

Worksheet answer

Q.1

(i) 9

(ii) 4

(iii) 6

(iv) 1

Q.2

(a) 2209      (b)1.5129

Q.3 14

Q.4

(a) No

(b) No

(c) Yes

Q.5

(a) 39

(b) 42

Q.6

Ans. 45m

Q.7

(i)2

(ii)3

Q.8

Ans.95

Q.9

Ans.100489

Q.10

Ans.998001

Q.11

Ans.18.2

Q.12

Ans.Rs.3600

Q.13

Ans.Rs. 150000

(II)  Fill in the blanks

(i) 100

(ii) ODD

(iii) 23

(III) True and false

(i) True

(ii) True

(iii) False

(iv) false

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