GRADE 8 Mathematics NUMBERS – Squares and Square Roots Notes
Mathematics — NUMBERS
Subtopic: Squares and Square Roots (Age: 13, Kenyan lower secondary)
What is a square?
The square of a number n means n multiplied by itself. We write it as n2 (read "n squared").
Examples:
42 = 4 × 4 = 16
72 = 49
(−3)2 = (−3) × (−3) = 9
42 = 4 × 4 = 16
72 = 49
(−3)2 = (−3) × (−3) = 9
Perfect squares (quick list)
Common perfect squares up to 152:
12=1
22=4
32=9
42=16
52=25
62=36
72=49
82=64
92=81
102=100
112=121
122=144
132=169
142=196
152=225
What is a square root?
The square root of a number x is a number y such that y2 = x. We use the symbol √ for the square root. The principal (or positive) square root is written as √x.
Example: √25 = 5 because 52 = 25.
Note: when solving equations like x2 = 25, x = ±5 (both +5 and −5 are solutions).
Note: when solving equations like x2 = 25, x = ±5 (both +5 and −5 are solutions).
How to find square roots (methods)
- Recognise perfect squares from the list. (fast for small numbers)
- Prime factorisation method: group pairs of equal prime factors.
Example (perfect): Find √144.
144 = 2×2 × 2×2 × 3×3 = 24 × 32. Take one of each pair: √144 = 22 × 3 = 4 × 3 = 12.
144 = 2×2 × 2×2 × 3×3 = 24 × 32. Take one of each pair: √144 = 22 × 3 = 4 × 3 = 12.
Example (simplify, not whole): √72 = √(36 × 2) = √36 × √2 = 6√2.
3) Estimate for non-perfect squares: find nearest smaller and larger perfect squares to get a rough decimal.
Important properties (simple)
- (ab)2 = a2 b2
- √(a × b) = √a × √b for non-negative a and b
- √(a/b) = √a / √b for b ≠ 0
- √(a2) = |a| (square root returns the non-negative value)
A quick practical link (Kenyan context)
If you have a square field with area 81 m2, the length of one side = √81 = 9 m. This idea helps when designing square gardens or tiles.
Worked examples
Example 1: √81 = 9 because 9×9 = 81.
Example 2: Simplify √180.
180 = 2×2 × 3×3 × 5 = 22 × 32 × 5 → √180 = 2×3×√5 = 6√5.
Example 2: Simplify √180.
180 = 2×2 × 3×3 × 5 = 22 × 32 × 5 → √180 = 2×3×√5 = 6√5.
Example 3 (estimation): Estimate √50.
72=49 and 82=64, so √50 is a little more than 7. Approx √50 ≈ 7.07.
72=49 and 82=64, so √50 is a little more than 7. Approx √50 ≈ 7.07.
Practice questions
- Find 112 and 122.
- Calculate √144 and √225.
- Simplify √200.
- Find √1 and √0.
- Simplify √72.
- Find the side of a square whose area is 196 m2.
- Use prime factors to find √360 (simplify).
- Estimate √200 to 2 decimal places (use nearest squares 142=196, 152=225).
Answers (check after trying)
1) 112 = 121, 122 = 144.
2) √144 = 12, √225 = 15.
3) √200 = √(100×2) = 10√2.
4) √1 = 1, √0 = 0.
5) √72 = √(36×2) = 6√2.
6) side = √196 = 14 m.
7) 360 = 2×2 × 2×3×3 ×5 = 23 × 32 × 5 → √360 = 2×3×√(2×5) = 6√10.
8) √200 ≈ √(196) = 14, √225 = 15 so √200 ≈ 14.14 (actual ≈ 14.1421).
2) √144 = 12, √225 = 15.
3) √200 = √(100×2) = 10√2.
4) √1 = 1, √0 = 0.
5) √72 = √(36×2) = 6√2.
6) side = √196 = 14 m.
7) 360 = 2×2 × 2×3×3 ×5 = 23 × 32 × 5 → √360 = 2×3×√(2×5) = 6√10.
8) √200 ≈ √(196) = 14, √225 = 15 so √200 ≈ 14.14 (actual ≈ 14.1421).