$$14^{\frac{5}{2}}\times42^{\frac{3}{2}}\times21^{\frac{3}{2}}\div7^{?}=432$$

Expression : $$14^{\frac{5}{2}}\times42^{\frac{3}{2}}\times21^{\frac{3}{2}}\div7^{?}=432$$

=> $$(2 \times 7)^{\frac{5}{2}} \times (2 \times 3 \times 7)^{\frac{3}{2}} \times (3 \times 7)^{\frac{3}{2}} \div (7)^{x} = 432$$

=> $$(7)^{\frac{5}{2} + \frac{3}{2} + \frac{3}{2}} \times (2)^{\frac{5}{2} + \frac{3}{2}} \times (3)^{\frac{3}{2} + \frac{3}{2}} \div (7)^x = 432$$

=> $$(7)^{\frac{11}{2}} \times (2)^4 \times (3)^3 \div (7)^x = 432$$

=> $$(7)^{\frac{11}{2} - x} = \frac{432}{16 \times 27}$$

=> $$(7)^{\frac{11}{2} - x} = 1 = (7)^0$$

=> $$\frac{11}{2} - x = 0$$

=> $$x = \frac{11}{2}$$