If $${\text{5}}\sqrt 5 \times {{\text{5}}^3} \div {{\text{5}}^{ - \frac{3}{2}}}{\text{ = }}{{\text{5}}^{a + 2}}{\text{,}}$$     then the value of a is = ?

$$\eqalign{ & {\text{5}}\sqrt 5 \times {{\text{5}}^3} \div {{\text{5}}^{ - \frac{3}{2}}}{\text{ = }}{{\text{5}}^{a + 2}} \cr & \Rightarrow {{\text{5}}^1} \times {5^{\frac{1}{2}}} \times {{\text{5}}^3} \div {{\text{5}}^{ - \frac{3}{2}}}{\text{ = }}{{\text{5}}^{a + 2}} \cr & \Rightarrow {5^{1 + \frac{1}{2} + 3 - \left( { - \frac{3}{2}} \right)}} = {5^{a + 2}} \cr & \Rightarrow {5^{1 + \frac{1}{2} + 3 + \frac{3}{2}}} = {5^{a + 2}} \cr & \Rightarrow {5^6} = {5^{a + 2}} \cr & \Rightarrow a + 2 = 6 \cr & \Rightarrow a = 4 \cr} $$