Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.34)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Laws of Exponents - Given an exponential equation of the form $x^{a/b} = c$, we can solve for $x^{1/b}$ by taking the appropriate roots on both sides. **2. Detailed Step-by-Step Calculation**: - **Given Equation**: $x^{3/7} = 216$ - **Recognizing prime powers**: $216 = 6^3$ $x^{3/7} = 6^3$ - **Take the cube root on both sides**: $(x^{3/7})^{1/3} = (6^3)^{1/3} \implies x^{1/7} = 6$ - **Find the required value $x^{-1/7}$**: $x^{-1/7} = \frac{1}{x^{1/7}} = \frac{1}{6} \approx 0.16667 \approx 0.167$ **3. Final Answer**: - **Calculated Value**: **0.167** - **Official GATE Accepted Range**: **0.166 to 0.168**