1.5 Squares and Square Roots
1.5.1 Squares
A square number is a number multiplied by itself.
6 × 6 = 36 or 6² = 36
Squares can be found for:
- whole numbers
- decimals
- fractions
23² = 529
7.9² = 62.41
(4 ÷ 7)² = 16 ÷ 49
1.5.2 Square Roots
The square root of a number is the inverse of squaring.
√36 = 6
Square roots of whole numbers, fractions, and decimals can be found using the factorisation method.
Example 1.32
Find √784
784 = 2 × 2 × 2 × 2 × 7 × 7
Pair equal factors:
(2 × 2) × (2 × 2) × (7 × 7)
Take one from each pair:
2 × 2 × 7 = 28
Pair equal factors:
(2 × 2) × (2 × 2) × (7 × 7)
Take one from each pair:
2 × 2 × 7 = 28
Example 1.33
Find √(1625 ÷ 1296)
√(1625 ÷ 1296) = √1625 ÷ √1296
1625 = 5 × 5 × 5 × 5 → √1625 = 25
1296 = 36 × 36 → √1296 = 36
Answer = 25 ÷ 36
1625 = 5 × 5 × 5 × 5 → √1625 = 25
1296 = 36 × 36 → √1296 = 36
Answer = 25 ÷ 36
Example 1.34
Find √20.25
20.25 = 2025 ÷ 100
√2025 = 45
√100 = 10
Answer = 45 ÷ 10 = 4.5
√2025 = 45
√100 = 10
Answer = 45 ÷ 10 = 4.5
Exercise 1.14
1. Solve each of the following by multiplication:
(a) 28²
28 × 28 = 784
(b) 2.56²
2.56 × 2.56 = 6.5536
(c) 63²
63 × 63 = 3969
(d) 0.81²
0.81 × 0.81 = 0.6561
2. Evaluate:
(a)
√3969
63 × 63 = 3969 → √3969 = 63
(b)
√47.61
6.9 × 6.9 = 47.61 → √47.61 = 6.9
(c)
√0.9216
0.96 × 0.96 = 0.9216 → √0.9216 = 0.96
3. Word Problems
3. The area of a square tile is 1024 cm².
Find the length of one side of the tile.
Area = side² → side = √1024 = 32 cm
4. A square blanket has a side length of 1.5 m.
Find the area of the blanket.
Area = side² = 1.5 × 1.5 = 2.25 m²
5. John has a square piece of land with an area of 5184 m².
Find the perimeter of the land.
Side = √5184 = 72 m
Perimeter = 4 × 72 = 288 m
Perimeter = 4 × 72 = 288 m