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Q. If \( a^{b^c} = 256 \), find maximum value of \( abc \)
(a) 12 (b) 16 (c) 32 (d) 256
✏️ Solution
\( 256 = 2^8 \)
Possible forms: \( a^{b^c} = 2^8 \)
Take \( a = 2 \Rightarrow b^c = 8 \)
Maximize \( bc \): \( 8 = 2^3 \Rightarrow b = 2,\ c = 3 \)
\( abc = 2 \cdot 2 \cdot 3 = 12 \)
Try better: \( a = 4 = 2^2 \Rightarrow (4)^{b^c} = 2^8 \Rightarrow b^c = 4 \)
\( 4 = 2^2 \Rightarrow b = 2,\ c = 2 \)
\( abc = 4 \cdot 2 \cdot 2 = 16 \)
Try \( a = 16 = 2^4 \Rightarrow b^c = 2 \)
\( b = 2,\ c = 1 \Rightarrow abc = 16 \cdot 2 \cdot 1 = 32 \)
Maximum = 32
Correct Option: (c) 32
\( \boxed{32} \)