$( \text{What approximate value will come in place of question mark (?) in the following questions?} )$ $( (99867)^{\frac{1}{2}} + (65547)^{\frac{1}{4}} + (19712)^{\frac{1}{3}} + (1314)^{\frac{1}{4}} = (6571)^{\frac{1}{4}} + ? )$

Approximating using roots: (99867)^(1/2) = √99867 ≈ 316, (65547)^(1/4) = √(√65547) ≈ √256 = 16, (19712)^(1/3) = ∛19712 ≈ 27, (1314)^(1/4) = √(√1314) ≈ √36 = 6. LHS = 316 + 16 + 27 + 6 = 365. RHS = (6571)^(1/4) + ? ≈ √(√6571) ≈ √81 = 9. So: 365 = 9 + ?, giving ? = 356. Option C is 356. Let me verify: √99867 = 315.99, ⁴√65547 = 15.97, ∛19712 = 27.02, ⁴√1314 = 6.01. LHS = 315.99 + 15.97 + 27.02 + 6.01 = 365. ⁴√6571 = 8.99. So: 365 = 8.99 + ?, ? = 356.01 ≈ 356. This confirms option C is correct.