Operations with Real Numbers — Solutions

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1. Add and subtract like surds

   1. (a) 3√2 + 5√2: 8√2
   2. (b) 7√3 − 2√3: 5√3
   3. (c) √8 + √18: 2√2 + 3√2 = 5√2
   4. (d) √12 + √27: 2√3 + 3√3 = 5√3
   5. (e) 2√50 − √8: 10√2 − 2√2 = 8√2
   6. (f) √75 + √48 − √27: 5√3 + 4√3 − 3√3 = 6√3
   7. (g) 3√20 − √45 + 2√5: 6√5 − 3√5 + 2√5 = 5√5
   8. (h) 2√72 + 3√32 − √50: 12√2 + 12√2 − 5√2 = 19√2

2. Multiply and divide surds

   1. (a) √3 × √3: 3
   2. (b) √5 × √20: √100 = 10
   3. (c) 2√3 × 4√3: 8 × 3 = 24
   4. (d) 3√2 × 5√8: 15√16 = 15 × 4 = 60
   5. (e) √6 × √10: √60 = √(4 × 15) = 2√15
   6. (f) √50 ÷ √2: √(50/2) = √25 = 5
   7. (g) √75 ÷ √3: √(75/3) = √25 = 5
   8. (h) 6√20 ÷ 3√5: (6/3) × √(20/5) = 2 × √4 = 2 × 2 = 4

3. Mixed operations

   1. (a) (√7)²: 7
   2. (b) (3√2)²: 9 × 2 = 18
   3. (c) 2√5 × 3√15: 6√75 = 6 × 5√3 = 30√3
   4. (d) (√3 + √3)²: (2√3)² = 4 × 3 = 12
   5. (e) √2 × √8 + √3 × √3: √16 + 3 = 4 + 3 = 7
   6. (f) 4√6 ÷ (2√2): (4/2) × √(6/2) = 2√3

4. True or False

   1. False. √2 ≈ 1.414, √3 ≈ 1.732; their sum ≈ 3.146. But √5 ≈ 2.236. The surds have different irrational parts — they cannot be added by adding the numbers under the roots.
   2. True. √4 × √9 = 2 × 3 = 6, and √36 = 6. Both are 6.
   3. False. 3√2 and 4√3 are unlike surds (different irrational parts). They cannot be combined by adding coefficients.
   4. True. √a × √a = (√a)² = a for any a ≥ 0.
   5. False. √50 = 5√2 and √2 = √2. 5√2 − √2 = 4√2, not √48. (You cannot subtract by subtracting numbers under the root.)
   6. True. √12 = √(4 × 3) = 2√3. So 2√3 = √12. ✓

5. Garden border

   1. √50 = √(25×2) = 5√2 m; √8 = √(4×2) = 2√2 m
   2. Perimeter = 2(5√2 + 2√2) = 2 × 7√2 = 14√2 m
   3. Area = √50 × √8 = √400 = 20 m² (rational). The product √50 × √8 = √400 = 20 because the surds are ‘conjugate’ in the sense that their product is a perfect square.

6. Tiling a floor

   1. Total length = 6 × √3 = 6√3 m
   2. 6√3 is already simplified (6√3 ≈ 10.39 m)
   3. Area = (√3)² = 3 m². This is rational, because √3 × √3 = 3.

7. Comparing surd expressions

   1. 3√2 vs 2√3: (3√2)² = 18; (2√3)² = 12. Since 18 > 12, 3√2 > 2√3
   2. 4√3 vs 3√5: (4√3)² = 48; (3√5)² = 45. Since 48 > 45, 4√3 > 3√5
   3. √8 + √2 vs √18: √8 + √2 = 2√2 + √2 = 3√2. √18 = 3√2. So √8 + √2 = √18

8. Rational or irrational?

   1. (a) √2 × √2 = 2. Rational
   2. (b) √2 + √2 = 2√2. Irrational (coefficient does not remove the surd)
   3. (c) √3 × √12 = √36 = 6. Rational
   4. (d) √5 + 1 is irrational: adding a rational (1) to an irrational (√5) gives an irrational. Irrational
   5. (e) (√5)² − 5 = 5 − 5 = 0. Rational

9. Pythagoras’ theorem

   1. c² = (√5)² + (√20)² = 5 + 20 = 25. c = √25 = 5 cm
   2. b² = 13 − (√4)² = 13 − 4 = 9. b = √9 = 3 cm
   3. d = s√2 ⇒ √72 = s√2 ⇒ s = √72/√2 = √36 = 6 cm

10. Surd investigation

    1. (√5 + √3)(√5 − √3) = (√5)² − (√3)² = 5 − 3 = 2. General rule: (√a + √b)(√a − √b) = a − b.
    2. (√7 + √2)(√7 − √2) = 7 − 2 = 5
    3. Counter-example: √4 + √9 = 2 + 3 = 5, but √(4+9) = √13 ≈ 3.6. Since 5 ≠ 3.6, the claim is false. The error is treating the square root as a linear function; √(a+b) ≠ √a + √b in general.