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

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Square and square root class 8 notes:[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: What is the mean of the square of 5. It is nothing but the multiplication of number to itself  5×5 =25.So,25 is our result.

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 see types of square

All Square is divided into two categories on the basis of its nature 

  • Perfect square
  • Non-perfect square

                                                  What is a perfect square?

A Square formed by multiply the whole number is called a perfect square.

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

Note:

(i)  All natural numbers are not perfect squares.

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

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

NUMBER SQUARE
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 will cover later. If you never read about square root. Don’t worry, I will explain this concept in the squares root lesson. But here is the definition of a non-perfect square.

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

Those who understood this, very good. But those not, don’t worry I will explain it later. Before that, let’s dive into the properties of the perfect square.

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 they ending with digit 2,3,7, and 8. But, it has some exception as well which you can found in the notes given below

Note: It is not necessary that a number ending with digit 0,1,4,5,6 or 9 will be a perfect square. For example, 184 is not a square number. 

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

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

Note: 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 .

Here is proof of above statement

Let n is any natural number. So, the next consecutive number will be (n+1)

(n+1)²-n²=(n+1+n)(n+1-n)

(n+1)²-n²=(n+1+n)(1)

(n+1)²-n²=(n+1)+(n)

Hence proved. Try to understand this concept with the help of an example

6²-5²=36-25=11

This difference can be found as 6+5=11

Now, you may be thinking “What is the need for the above statement, we can easily found it by subtracting?” Right

I ask you a question. Find 79²-78²

I think “you are thinking about calculating the square of both numbers  ” right

But it is a sum of 79+78=157

I think now you have understood about above concept .ok

Here are some other post for class 8 which might be helpful for your audience

6. Pythagorean triplet

What is the Pythagorean triplet?

A Pythagorean triplet is three number which satisfies Pythagoras theorem. For example, 3,4 and 5 are Pythagorean triplet

because

5²=3²+4²

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

Through the above formula, you can find many triplets you want but keep in mind “Value of n should be greater than 1”.The smallest Pythagorean triplet is given above 3,4 and 5 which you can calculate by substituting n=2. So, all the numbers which are Pythagorean triplets also are perfect squares.

7.On dividing a perfect square by 3, either 0 or 1 is left as a reminder

Example:

4=3×1+1

9=3×3+0

16=3×5+1  ….. so on

Here, you can see the remainder are either 0 or 1

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 squares

9. Here is an amazing relation between perfect squares and square of digit of series one

         1²=1
        11²=121 , 1+2+1=4=2²
       111²=12321 ,1+2+3+2+1=3²
             ...
            .....

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

I think you notice relations between numbers

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

Here is an example: 10,11,12,13,14,15 numbers are lying between 9 and 16 which are perfect squares

The value of n =3

The number lying between 3 and 4 should be 6, which you see in the above sequence is true.

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.

As I promised to explain the concept of an imperfect square. So, here is an explanation

When the square of root of a number is not whole number(it may in a decimal form) ,that number is called non-perfect square .For example,

√5=2.73 approx

So, 5 is  known as imperfect square

……

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 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

Q. Find the square root of 784

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

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

How to find square root using the ones and tens method

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

1 ⇒ 1 or 9 
4 ⇒ 2 or 8 
6 ⇒ 4 or 6 
9 ⇒ 3 or 7 
5 ⇒ 5 
0 ⇒ 0

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

Enough concept!

Let’s understand with the help of an example

Q. 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 digit 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 this 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

How to find square root using long -division method

First, Let’s learn about determining the number of digits in the square root of a given number.

Simply put the bar over every pair of digits starting from the rightmost or from unit place

Note: Number of bars=Number of digits in the square root of the given number

For example:

Q. Determine the number of digits in the square root of 2704

Sol. \( \overline{27}\)  \( \overlinne{04}\)

Number of digits in the square root of 2704=2(Number of bars =2)

Note:

1 If a number has odd number of digits ,then number of digits in the square root will be \frac{n+1}{2}

2 If number of digits are even ,then number of digits in their square root will be \frac{n}{2}

Let’s see how we can find the square root of a number using the long division method

Q.1 Find the square root of 1024 using the long division method

Sol.square root using long division method

image credit=examfear.com

In this way, we can easily calculate the square root of any number

Square and square root class 8 notes :[Word problem solution]

Q.1 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.

Q.2 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

Q.3 Find the smallest 4-digits number which is a perfect square

Solution:

We know, smallest 4-digits number=1000

24 should be added to 1000 to obtain the smallest 4-digit perfect square. Smallest 4-digits perfect square is 1000+24=1024

NOTE: We cannot subtract a number from 1000 as on subtracting we obtain a 3-digit 
number and our requirement is of a 4-digit number .

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 read earlier

(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 lies 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

Square and the square root class 8 :[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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