Exercise 6.2

1. Find the square root of the following numbers by Prime factorization method.

(i) 441

Solution:

Prime factors of 441: × × ×

Pair the prime factors: (3 × 3) × (7 × 7)

Take one factor from each pair: 3 × 7

Multiply the factors: 3 × 7 =

Therefore, the square root of 441 is .

(ii) 784

Solution:

Prime factors of 784: × × × × ×

Pair the prime factors: (2 × 2) × (2 × 2) × (7 × 7)

Take one factor from each pair: 2 × 2 × 7

Multiply the factors: 2 × 2 × 7 =

Therefore, the square root of 784 is .

(iii) 4096

Solution:

Prime factors of 4096: × × × × × × × × × × ×

Pair the prime factors: (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)

Take one factor from each pair: 2 × 2 × 2 × 2 × 2 × 2

Multiply the factors: 2 × 2 × 2 × 2 × 2 × 2 =

Therefore, the square root of 4096 is .

(iv) 7056

Solution:

Prime factors of 7056: × × × × × × ×

Pair the prime factors: (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7)

Take one factor from each pair: 2 × 2 × 3 × 7

Multiply the factors: 2 × 2 × 3 × 7 =

Therefore, the square root of 7056 is .

2. Find the smallest number by which 3645 must be multiplied to get a perfect square.

Solution:

Prime factorize 3645: 3536 × 5152

To make it a perfect square, each prime factor must have an power. We need one more 3 and one more 5.

The smallest number to multiply is 3 × 5 = .

3. Find the smallest number by which 2400 is to be multiplied to get a perfect square and also find the square root of the resulting number.

Solution:

Prime factorize 2400: 2524 × 3132 × 52153

To make it a perfect square, we need one more 2 and one more 3.

The smallest number to multiply is 2 × 3 = .

The resulting number is 2400 × 6 = = 2624 × 3234 × 5254

The square root of 14400 is 23 × 3 × 5 = 8 × 3 × 5 = .

4. Find the smallest number by which 7776 is to be divided to get a perfect square.

Solution:

Prime factorize 7776: 7776 = 2524 × 3534

To make it a perfect square by division, we need to remove the extra factors so that all powers are .

We can divide by 2 × 3 = .

The smallest number to divide by is .

5. 1521 trees are planted in a garden in such a way that there are as many trees in each row as there are rows in the garden. Find the number of rows and number of trees in each row.

Solution:

Let 'x' be the number of rows and also the number of trees in each row.

Total trees = x × x = x2 = 1521

Find the square root of 1521:

1521 = 3234 × 132133 = 39239

x = .

Therefore, there are rows and trees in each row.

6. A school collected ₹2601 as fees from its students. If fee paid by each student and number students in the school were equal, how many students were there in the school?

Solution:

Let 'x' be the number of students and also the fee paid by each student.

Total fee collected = × = x2x4 = ₹

Find the square root of 2601:

2601 = 3254 × 172194 = 3×172 = 512492

x = .

Therefore, there were 51 students in the school.

7. The product of two numbers is 1296. If one number is 16 times the other, find the two numbers?

Solution:

Let one number be 'x'. Then the other number is .

Product = x × 16x = 16x232x = 1296

x2 = 129616 =

x = 81 =

The two numbers are and 16 × 9 = .

8. 7921 soldiers sat in an auditorium in such a way that there are as many soldiers in a row as there are rows in the auditorium. How many rows are there in the auditorium?

Solution:

Let 'x' be the number of rows and also the number of soldiers in each row.

Total soldiers = x × x = x2 = 7921

Find the square root of 7921: 7921 = 892792

x = .

Therefore, there are 89 rows in the auditorium.

9. The area of a square field is 5184 m². Find the area of a rectangular field, whose perimeter is equal to the perimeter of the square field and whose length is twice of its breadth.

Solution:

Side of the square field = 5184 = m.

Perimeter of the square field = × 72 = m.

Let the breadth of the rectangular field be 'b'. Then the length is .

Perimeter of the rectangular field = 2(l + b) = 2( + b) = = m.

b = 2886 = m.

Length of the rectangular field = × 48 = m.

Area of the rectangular field = length × breadth = 96 × 48 = m2.