If \( 12 ^ { 12 x } \times 4 ^ { 24 x + 12 } \times 5 ^ { 2 y } = 8 ^ { 4 z } \times 20 ^ { 12 x } \times 243 ^ { 3 x - 6 } \), where \( x , y \) and \( z \) are natural numbers, then \( x + y + z \) equals
Answer: 112
Key idea: Break every base into primes and equate the powers of 2, 3 and 5.
1. LHS: \( 12^{12x} = 2^{24x}3^{12x} \), \( 4^{24x+12} = 2^{48x+24} \), so LHS \( = 2^{72x+24} \cdot 3^{12x} \cdot 5^{2y} \).
2. RHS: \( 8^{4z} = 2^{12z} \), \( 20^{12x} = 2^{24x}5^{12x} \), \( 243^{3x-6} = 3^{15x-30} \), so RHS \( = 2^{12z+24x} \cdot 3^{15x-30} \cdot 5^{12x} \).
3. Powers of 3: \( 12x = 15x - 30 \Rightarrow x = 10 \).
4. Powers of 5: \( 2y = 12x = 120 \Rightarrow y = 60 \).
5. Powers of 2: \( 72x + 24 = 12z + 24x \Rightarrow 744 = 12z + 240 \Rightarrow z = 42 \).
So \( x + y + z = 10 + 60 + 42 = 112 \).
Hence, the answer is 112.