The Worksheet on Squares is useful for students to prepare for their examinations. In this worksheet to calculate squares of the numbers, we will multiply the number by itself and the square root of a number is a value that can be multiplied by itself and to get the original value. If we have a negative number to calculate the square then by multiplying a negative number by itself then the number will be a positive number.

And here to calculate perfect squares by using the prime factorization method we will find the prime factorization of that number and then we will divide the prime factorization into equal pairs of factors by which we can get the perfect squares. And to find the squares of even numbers we can check the last digit of the number, if the number is odd then that number will never be the square of the even number. In the below, we can see the examples with the solutions.

**1.** Find which of the following numbers are perfect squares by using the prime factorization method?

(i) 343

(ii) 2048

## Solution:

(i) 81

81= (3×3)×(3×3)

As it has equal pairs of factors.

So it is a perfect square.

(ii) 2048

2048=(2×2)×(2×2)×(2×2)×(2×2)×(2×2)×2

As 2048 does not have any equal pairs of factors.

So it is not a perfect square.

**2.** Give reason in each case and show that none of the following is a perfect square.

(i) 189

(ii)3549

## Solution:

(i) 189

189=3^2×3×7

As the number 189 does not have equal pairs of factors it is not a perfect square.

(ii)3549

3549=(13×13)×3×7

As the number 3549 does not have equal pairs of factors it is not a perfect square.

**3.** Show that the following are the squares of even numbers?

(i) 784

(ii) 144

## Solution:

(i) 784

784: 28^2 as 28 is an even number.

(ii) 144

144: 12^2 as 12 is an even number.

**4.** Show that the following are the squares of odd numbers?

(i) 169

(ii) 289

## Solution:

(i) 169

169: 13^2 as 13 is an odd number.

(ii) 289

289: 17^2 as 17 is an odd number.

**5.** Calculate:

(i) (48)² – (47)²

(ii) (57)² – (56)²

## Solution:

(i) (48)² – (47)²

= 48+47

= 95

(ii) (57)² – (56)²

= 57+56

= 113

**6.** Find the sum without adding:

(i) (1 + 3 + 5 + 7 + 9 + 11 + 13)

(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17)

## Solution:

(i) (1 + 3 + 5 + 7 + 9 + 11 + 13)

As the total consecutive odd numbers are 7

and n = 7

Therefore sum = n×n

= 7×7

= 49

(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17)

As the total consecutive odd numbers are 9

and n = 9

Therefore sum = n×n

= 9×9

= 81

**7.** Determine 32 as the sum of 4 odd numbers.

## Solution:

x+x+2+x+4+x+6=32

4x+12=32

4x= 32-12

4x= 20

x= 5

5,7,9,11= 32

**8.** Write a Pythagorean triplet whose smallest member is 8?

## Solution:

The Pythagorean triplet is 2m, m^2 – 1 and m^2+1

and the smallest member is 8,

So

2m= 8

m= 8/2

= 4.

As m= 4

2m= 2×4

= 8.

m^2-1= (4)^2 – 1

= 16 – 1

= 15.

m^2+1= (4)^2 + 1

= 16+1

= 17.

So, the Pythagorean triplet is 8,15,17.

**9.** Prove that 4,520 is not a perfect square.

## Solution:

To get the perfect square the number should have an equal pair of factors, so it will be a perfect square.

Here the number is 4,520 and the prime factorization of the number 4,520 is

4,520= 2×2×2×5×113

Here the number 4,520 didn’t have a perfect square, so the number 4,520 is not a perfect square.

**10.** Is the number 15,625 a perfect square or not?

## Solution:

The prime factorization of 15,625 is

15,625= 5×5×5×5×5×5

which has an equal pair of factors. So the number 15,625 is a perfect square.

**11.** Write Yes or No for each of the statements given below:

(i) For a perfect square, the number of digits should be even?

(ii) Do the square of a prime number is also prime?

## Solution:

(i) False.

As a perfect square number will also be an odd number.

For example:

169= 13×13

As 13 is an odd number.

(ii)False.

As the square of the prime number can be an even number, an odd number, or a composite number.

For example:

The square of the number 3 is 3×3= 9 which is an odd number.