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Mathematics

If 2a - 3b = 3 and ab = 2, find the value of 8a3 - 27b3.

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Answer

We know that,

⇒ (a - b)3 = a3 - b3 - 3ab(a - b).

∴ a3 - b3 = (a - b)3 + 3ab(a - b).

∴ 8a3 - 27b3 = (2a)3 - (3b)3 = (2a - 3b)3 + 3 × 2a × 3b (2a - 3b)

Substituting values we get,

⇒ 8a3 - 27b3 = (2a - 3b)3 + 3 × 2a × 3b (2a - 3b)

⇒ 8a3 - 27b3 = 33 + 18ab × 3

⇒ 8a3 - 27b3 = 27 + 18 × 2 × 3

⇒ 8a3 - 27b3 = 27 + 108

⇒ 8a3 - 27b3 = 135.

Hence, 8a3 - 27b3 = 135.

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