If \(X^{y^z}=1, Y^{z^x}=125\) and \(Z^{y^x}=243\) (x, y and z are natural numbers), then what is the value of 9x + 10y - 18z?
5
Step 1 — From \(X^{y^z}=1\): since x, y, z are natural, the exponent \(y^z\ge 1\), so \(X=1\).
Step 2 — From \(Y^{z^x}=125=5^3\): take \(Y=5,\ z^x=3\Rightarrow z=3,\ x=1\).
Step 3 — From \(Z^{y^x}=243=3^5\): \(Z=3,\ y^x=5\Rightarrow y=5,\ x=1\). Consistent.
Step 4: \(9(1)+10(5)-18(3)=9+50-54=\textbf{5}\).
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