Calculate $7$ to the power of $3$ and get $343$.

Calculate $5$ to the power of $343$ and get $558099312149548331160050560695967265357377927159569658796448165638492498001337009607152354423912123852140163258483412222763936212052694702698262838579121700178473641296916631351785077979957044026185393470473883326121722348034381866455078125$.

Factor $558099312149548331160050560695967265357377927159569658796448165638492498001337009607152354423912123852140163258483412222763936212052694702698262838579121700178473641296916631351785077979957044026185393470473883326121722348034381866455078125=334095588761524455767567058393935234851899610013131087245242186520049830123391294833812281694918056018650531768798828125^{2}\times 5$. Rewrite the square root of the product $\sqrt{334095588761524455767567058393935234851899610013131087245242186520049830123391294833812281694918056018650531768798828125^{2}\times 5}$ as the product of square roots $\sqrt{334095588761524455767567058393935234851899610013131087245242186520049830123391294833812281694918056018650531768798828125^{2}}\sqrt{5}$. Take the square root of $334095588761524455767567058393935234851899610013131087245242186520049830123391294833812281694918056018650531768798828125^{2}$.

Multiply $\frac{3}{2}$ and $334095588761524455767567058393935234851899610013131087245242186520049830123391294833812281694918056018650531768798828125$ to get $\frac{1002286766284573367302701175181805704555698830039393261735726559560149490370173884501436845084754168055951595306396484375}{2}$.