$$5 \sqrt { 5 } \times 5 ^ { 3 } \div 5 ^ { - \frac { 3 } { 2 } } = 5 ^ { a + 2 }$$
$a=4$
$$\frac{5^{4}\sqrt{5}}{5^{-\frac{3}{2}}}=5^{a+2}$$
$$\sqrt{5}\times 5^{\frac{11}{2}}=5^{a+2}$$
$$5^{a+2}=\sqrt{5}\times 5^{\frac{11}{2}}$$
$$5^{a+2}=3125\sqrt{5}\sqrt{5}$$
$$\log(5^{a+2})=\log(15625)$$
$$\left(a+2\right)\log(5)=\log(15625)$$
$$a+2=\frac{\log(15625)}{\log(5)}$$
$$a+2=\log_{5}\left(15625\right)$$
$$a=6-2$$
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