ICSE Class 8 Maths Chapter 05 Squares and Square Roots

Chapter 5: Squares And Square Roots

5.1 Review

4² = 16 is also read as; 4 raised to the power 2 is 16.

Squares of even numbers are always even.
2² = 2 × 2 = 4; 6² = 6 × 6 = 36; 14² = 14 × 14 = 196 and so on.

Squares of odd numbers are always odd.
3² = 3 × 3 = 9; 7² = 49; 15² = 225 and so on.

Whether the number is negative or positive, its square is always positive.
(3)² = 3 × 3 = 9, which is a positive number.
(-3)² = -3 × -3 = 9, which is also a positive number.
Similarly, (-5)² = 25 and (5)² = 25, (-8)² = 64 and 8² = 64.

Since, the square of every number is positive, the square root of a positive number can be obtained, but the square root of a negative number is not possible.

Test Yourself

1. The square of 0.5 = ......... and the square root of 0.49 = ..........

2. The square root of 81 = ......... and the square of 25 = ..........

3. Square of an odd number is always an ......... number.

4. Square of an even number is always an ......... number.

5. Square of every integer is ..........

6. Square of every negative number is ..........

Teacher's Note

When you arrange items in a square grid, like arranging books on a 5 x 5 shelf, the total number of items is a perfect square. Understanding squares helps in real-world arrangements and calculations.

5.2 Perfect Square

A number, whose exact square root can be obtained, is called a perfect square.

16, 49, 1.21, \(\frac{9}{16}\), etc are perfect squares as \(\sqrt{16}\) = 4, \(\sqrt{49}\) = 7, \(\sqrt{1.21}\) = 1.1 and so on.

To find out whether a given number is a perfect square or not, express the number as a product of its prime factors. If the number is a perfect square, you would be able to group all the factors in pairs in such a way that both the factors in each pair are equal.

Example 1

Is 196 a perfect square?

196 = 2 × 2 × 7 × 7 = 2² × 7²

∴ The prime factors of 196 can be grouped in pairs; 196 is a perfect square.

Example 2

Is 180 a perfect square?

180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5

Since, all the prime factors of 180 cannot be grouped in pairs. [One factor (i.e. 5) is left]

∴ 180 is not a perfect square.

Teacher's Note

Perfect squares appear frequently in construction and tiling projects. For example, knowing that 144 is a perfect square (12 × 12) helps calculate square tiles needed for a floor.

5.3 To Find The Square Root Of A Perfect Square Number (Using Prime Factor Method)

Example 3

Find the square root of 484.

Square root of 484 = \(\sqrt{484}\)

Steps:
1. Resolve the number into prime factors : = \(\sqrt{2 \times 2 \times 11 \times 11}\)
2. Make pairs such that both the factors in each pair are equal : = \(\sqrt{(2 \times 2) \times (11 \times 11)}\)
3. Take one factor from each pair : = 2 × 11
4. The product is the square root of the given number = 22

Example 4

Find the smallest number by which 980 be multiplied so that the product is a perfect square.

980 = 2 × 2 × 5 × 7 × 7

Since, the prime factor 5 is not in pair.

∴ The given number should be multiplied by 5.

980 × 5 = 2 × 2 × 5 × 5 × 7 × 7, ∴ \(\sqrt{980 \times 5}\) = 2 × 5 × 7 = 70

Example 5

Find the smallest number by which 3150 be divided, so that the quotient is a perfect square.

3150 = 2 × 5 × 5 × 3 × 3 × 7

Since, the prime factors 2 and 7 cannot be paired.

∴ The given number should be divided by 2 × 7 = 14

\(\frac{3150}{14}\) = \(\frac{2 \times 5 \times 5 \times 3 \times 3 \times 7}{2 \times 7}\) = 5 × 5 × 3 × 3

Teacher's Note

Manufacturing and packaging often requires perfect square dimensions. Understanding how to find and use perfect squares helps in efficient production planning.

Example 6

(i) Find the square root of: \(2\frac{7}{9}\)

(ii) 4.41

Square root of \(2\frac{7}{9}\) = \(\sqrt{\frac{25}{9}}\) = \(\frac{5}{3}\) = \(1\frac{2}{3}\)

\(\sqrt{4.41}\) = \(\sqrt{\frac{441}{100}}\) = \(\frac{\sqrt{3 \times 3 \times 7 \times 7}}{\sqrt{2 \times 2 \times 5 \times 5}}\) = \(\frac{3 \times 7}{2 \times 5}\) = \(\frac{21}{10}\) = 2.1

Square root of a fraction = \(\frac{\text{Square root of its numerator}}{\text{Square root of its denominator}}\)

Instead of writing the prime factors of the given number in pairs, we can write them in index form and then in order to find the required square root, take half of each index value.

\(\sqrt{784}\) = \(\sqrt{2 \times 2 \times 2 \times 2 \times 7 \times 7}\) = \(\sqrt{2^4 \times 7^2}\) = 2² × 7¹ = 28

\(\sqrt{9}\) = 3, but \(\sqrt{0.9}\) ≠ 0.3

Reason: (0.3)² = 0.3 × 0.3 = 0.09 ∴ \(\sqrt{0.09}\) = 0.3

In the same way, \(\sqrt{144}\) = 12, but \(\sqrt{14.4}\) ≠ 1.2

Reason: (1.2)² = 1.2 × 1.2 = 1.44 ∴ \(\sqrt{1.44}\) = 1.2

Square root of a perfect square even number is always an even number and square root of a perfect square odd number is always an odd number.

(i) \(\sqrt{4}\) = 2, \(\sqrt{16}\) = 4, \(\sqrt{36}\) = 6, \(\sqrt{64}\) = 8, \(\sqrt{100}\) = 10 and so on.

(ii) \(\sqrt{9}\) = 3, \(\sqrt{25}\) = 5, \(\sqrt{49}\) = 7, \(\sqrt{81}\) = 9, \(\sqrt{121}\) = 11 and so on.

Example 7

A man plants his orchard with 5625 trees and arranges them so that there are as many rows as there are trees in each row. How many rows are there?

Let the number of rows be x.
∴ Number of trees in each row = x
and, total number of trees planted = x × x = x²

Given: x² = 5625 => x = \(\sqrt{5625}\) = \(\sqrt{5 \times 5 \times 5 \times 5 \times 3 \times 3}\) = 5 × 5 × 3 = 75

∴ The number of rows = 75

Example 8

In a basket there are 50 flowers. A man goes to worship and puts as many flowers in each temple as there are temples in the city. Thus, he needs 8 baskets of flowers. Find the number of temples in the city.

Let the number of temples in the city = x
∴ The number of flowers put in each temple = x

and, the total number of flowers used = x × x = x²
According to the given statement:
x² = 50 × 8 => x = \(\sqrt{50 \times 8}\) = \(\sqrt{5 \times 5 \times 2 \times 2 \times 2 \times 2}\) = 5 × 2 × 2 = 20

∴ The number of temples in the city = 20

Teacher's Note

Square root applications appear in agricultural planning when arranging crops in square fields and in event planning when organizing seating in square configurations.

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