A perfect square is a number that can be expressed as the product of an integer multiplied by itself. Numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 are perfect squares.

Method 1: Using Multiplication Facts

For small perfect squares, you can use multiplication tables you already know.

Example 1: Find √36

• Think: What number multiplied by itself equals 36?
• 6 × 6 = 36, so √36 = 6

Example 2: Find √81

• Think: What number multiplied by itself equals 81?
• 9 × 9 = 81, so √81 = 9

Method 2: Prime Factorization

When dealing with larger numbers, we can break them down into prime factors.

Steps:

1. Find the prime factors of the number
2. Group the factors into pairs
3. Take one factor from each pair and multiply them together

Example: Find √144

Step 1: Find prime factors of 144

2 | 144
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
  | 1

So 144 = 2 × 2 × 2 × 2 × 3 × 3

Step 2: Group into pairs

144 = (2 × 2) × (2 × 2) × (3 × 3)

Step 3: Take one factor from each pair and multiply

√144 = 2 × 2 × 3 = 12

Therefore, √144 = 12 (because 12 × 12 = 144)

Example 1: Find √64

• Using multiplication: 8 × 8 = 64, so √64 = 8
• Using prime factors: 64 = 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2) × (2 × 2) × (2 × 2)
• Taking one from each pair: 2 × 2 × 2 = 8

Example 2: Find √100

• 10 × 10 = 100, so √100 = 10

Example 3: Find √169

• Prime factors: 169 = 13 × 13
• √169 = 13 (because 13 × 13 = 169)

In Tanzania, square roots are useful when calculating areas. For example, if a farmer's rectangular plot of land has an area of 400 square metres and one side is 20 metres, they can use square root (√400 = 20) to find that the other side is also 20 metres, confirming the plot is square-shaped. This helps in planning how much fertilizer or seeds are needed for the area.