Question 11 of 29advanced🔍 AnalyzeNumerical3 marks

Which of the following is the greatest? Explain your reasoning. (i) 67³ - 66³ (ii) 43³ - 42³ (iii) 67² - 66² (iv) 43² - 42²

Correct Answer

Formula: a³ - b³ = (a-b)(a² + ab + b²) a² - b² = (a-b)(a+b)

(i) 67³ - 66³: 67³ - 66³ = (67-66)(67² + 67×6667 \times 66 + 66²) = 1 × (4489 + 4422 + 4356) = 13267

(ii) 43³ - 42³: 43³ - 42³ = (43-42)(43² + 43×4243 \times 42 + 42²) = 1 × (1849 + 1806 + 1764) = 5419

(iii) 67² - 66²: 67² - 66² = (67-66)(67+66) = 1×1331 \times 133 = 133

(iv) 43² - 42²: 43² - 42² = (43-42)(43+42) = 1×851 \times 85 = 85

Comparison: 13267 > 5419 > 133 > 85

The cube differences are much larger than the square differences, and 67³ - 66³ > 43³ - 42³.

Answer: (i) 67³ - 66³ is the greatest.

Exercise: Figure it Out (Section 2) | Q: 5 | (Page Page 16)
For More Understanding

Explanation

The chapter discusses differences between consecutive squares and cubes. For squares: a² - b² = (a-b)(a+b). For cubes: a³ - b³ = (a-b)(a² + ab + b²). When a and b differ by 1, the cube difference is much larger than the square difference for the same numbers.

Solution Steps

  1. Step 1: Calculate 67³ - 66³ = (67-66)(67² + 67×6667 \times 66 + 66²) = 1 × (4489 + 4422 + 4356) = 13267

  2. Step 2: Calculate 43³ - 42³ = (43-42)(43² + 43×4243 \times 42 + 42²) = 1 × (1849 + 1806 + 1764) = 5419

  3. Step 3: Calculate 67² - 66² = (67-66)(67+66) = 1×1331 \times 133 = 133

  4. Step 4: Calculate 43² - 42² = (43-42)(43+42) = 1×851 \times 85 = 85

  5. Step 5: Compare: 13267 > 5419 > 133 > 85. Greatest is 67³ - 66³

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