Simplifying Surds — Solutions

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1. Simplify each surd

   1. (a) √8: √(4×2) = 2√2
   2. (b) √18: √(9×2) = 3√2
   3. (c) √50: √(25×2) = 5√2
   4. (d) √75: √(25×3) = 5√3
   5. (e) √12: √(4×3) = 2√3
   6. (f) √45: √(9×5) = 3√5
   7. (g) √72: √(36×2) = 6√2
   8. (h) √98: √(49×2) = 7√2

2. Simplify with coefficients

   1. (a) 2√12: 2×2√3 = 4√3
   2. (b) 3√20: 3×2√5 = 6√5
   3. (c) 5√8: 5×2√2 = 10√2
   4. (d) 4√27: 4×3√3 = 12√3
   5. (e) 2√45: 2×3√5 = 6√5
   6. (f) 3√50: 3×5√2 = 15√2
   7. (g) ½√48: ½×4√3 = 2√3
   8. (h) 6√75: 6×5√3 = 30√3

3. Expand and simplify

   1. (a) √2(3 + √2): 3√2 + (√2)² = 3√2 + 2 = 2 + 3√2
   2. (b) √3(2√3 − √12): 2×3 − √36 = 6 − 6 = 0
   3. (c) 2√5(3√5 + √20): 2√5×3√5 + 2√5×2√5 = 30 + 20 = 50
   4. (d) (√6 + 2)(√6 − 2): (√6)² − 4 = 6 − 4 = 2

4. True or False

   1. True. √20 = √(4×5) = 2√5. ✓
   2. False. √8 = 2√2, so √8 + √2 = 2√2 + √2 = 3√2, not √10. You cannot add surds by adding the numbers under the root.
   3. True. √18 = √(9×2) = 3√2. ✓
   4. False. Counter-example: √(9+16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The square root of a sum is not equal to the sum of the square roots.
   5. True. √5 + √5 = 2√5. (2√5)² = 4×5 = 20. ✓
   6. True. √200 = √(100×2) = 10√2. ✓

5. Photo frame

   1. √50 = 5√2, √8 = 2√2. Perimeter = 2(5√2 + 2√2) = 2×7√2 = 14√2 cm
   2. Area = √50 × √8 = √400 = 20 cm² (whole number). The product √50 × √8 = √400 is a perfect square, making it rational.
   3. Frame 2: perimeter = 2(3√2 + 2√2) = 2×5√2 = 10√2 cm; area = 3√2 × 2√2 = 6×2 = 12 cm². Frame 2 is smaller (shorter sides: 3√2 ≈ 4.24 vs 5√2 ≈ 7.07 for Frame 1).

6. Equilateral triangle

   1. Perimeter = 3√12 = 3×2√3 = 6√3 cm
   2. Area = (√3/4)×(√12)² = (√3/4)×12 = 3√3 cm²
   3. 6√3 ≈ 6 × 1.732 = 10.392 ≈ 10.4 cm. ✓

7. Square patch

   1. Side = √48 = √(16×3) = 4√3 m
   2. Perimeter = 4 × 4√3 = 16√3 m
   3. The perimeter is irrational: 16√3 cannot be written as p/q because √3 is irrational, and 16 × (irrational) remains irrational.

8. Compare two squares

   Square A side = √18 = 3√2 m  Square B side = √50 = 5√2 m
   Square A perimeter = 4×3√2 = 12√2 m  Square B perimeter = 4×5√2 = 20√2 m
   Ratio: Side B / Side A = 5√2 / 3√2 = 5/3. Also √(50/18) = √(25/9) = 5/3. ✓

9. Pythagoras’ theorem

   1. c² = (√5)² + (√20)² = 5 + 20 = 25. c = √25 = 5 cm
   2. c² = (2√3)² + (√3)² = 12 + 3 = 15. c = √15 cm
   3. b² = (√108)² − 6² = 108 − 36 = 72. b = √72 = √(36×2) = 6√2 cm

10. Surd spiral

    1. The pattern: hypotenuse of nth triangle = √(n+1). So triangle 4 = √5, triangle 5 = √6, triangle 6 = √7.
    2. √8 = 2√2 (triangle 7, hypotenuse √8). √12 = 2√3 (triangle 11, hypotenuse √12).
    3. The hypotenuse of triangle n is √(n+1). For this to be rational, (n+1) must be a perfect square. The first perfect square after 1 is 4, so n+1=4, n=3. The 3rd triangle (with legs √3 and 1) has hypotenuse √4 = 2 (rational).