Prove that $(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})^{\frac{1}{2}} = (1^{3}+2^{3}+3^{3}+4^{3})^{\frac{1}{2}} + (5^{3})^{\frac{1}{3}}$.

(N/A) Step $1$: Calculate the sum inside the parentheses on the Left Hand Side $(LHS)$:
$1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} = 1 + 8 + 27 + 64 + 125 = 225$.
Step $2$: Evaluate the $LHS$:
$(225)^{\frac{1}{2}} = \sqrt{225} = 15$.
Step $3$: Calculate the sum inside the first parentheses on the Right Hand Side $(RHS)$:
$1^{3} + 2^{3} + 3^{3} + 4^{3} = 1 + 8 + 27 + 64 = 100$.
Step $4$: Evaluate the $RHS$:
$(100)^{\frac{1}{2}} + (5^{3})^{\frac{1}{3}} = \sqrt{100} + 5^{(3 \times \frac{1}{3})} = 10 + 5^{1} = 10 + 5 = 15$.
Step $5$: Since $LHS = 15$ and $RHS = 15$,the equation is proven.