Question
Download Solution PDFIf 8a3 + 27b3 = 16 and 2a + 3b = 4, then find value of 16a4 + 81b4.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
8a3 + 27b3 = 16 and 2a + 3b = 4
Concept used:
1. (A + B)2 = A2 + B2 + 2AB
2. (A + B)3 = A3 + B3 + 3AB(A + B)
Calculation:
8a3 + 27b3 = 16 ....(1)
⇒ (2a + 3b)3 - 18ab(2a + 3b) = 16
⇒ 43 - 18ab × 4 = 16
⇒ 72ab = 64 - 16
⇒ 72ab = 48
⇒ ab = 2/3 ....(2)
(2a + 3b) = 4 ....(3)
⇒ 4a2 + 12ab + 9b2 = 16
⇒ 4a2 + 9b2 = 16 - 12 × (2/3)
⇒ 4a2 + 9b2 = 8 ....(3)
Now,
(8a3 + 27b3)(2a + 3b) = 64
⇒ 16a4 + 54ab3 + 24a3b + 81b4 = 64
⇒ 16a4 + 81b4 + 6ab(4a2 + 9b2) = 64
⇒ 16a4 + 81b4 + 6 × (2/3) × 8 = 64
⇒ 16a4 + 81b4 = 64 - 32
⇒ 16a4 + 81b4 = 32
∴ The required value is 32.
Last updated on Jul 19, 2025
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