Question

​If 5√5 * 5³ / 5^-3/2 = 5^a+2, then what is the value of a?

Answer 1

$$\begin{gathered} \frac{5\sqrt{5}\times 5^3}{5^{-3/2}}=5^{\mathrm{a}+2} \\ \Rightarrow \frac{5\times 5^{1/2}\times 5^3}{5^{-3/2}}=5^{\mathrm{a}+2} \\ \Rightarrow \frac{5^{1+{\displaystyle \frac{1}{2}}+3}}{5^{-3/2}}=5^{\mathrm{a}+2} \\ \Rightarrow \frac{5^{9/2}}{5^{-3/2}}=5^{\mathrm{a}+2} \\ \Rightarrow 5^{\frac{9}{2}-\left(-\frac{3}{2}\right)}=5^{\mathrm{a}+2} \\ \Rightarrow 5^{\frac{9+3}{2}}=5^{\mathrm{a}+2} \\ \Rightarrow 5^6=5^{\mathrm{a}+2} \\ \Rightarrow 6=\mathrm{a}+2 \\ \Rightarrow \mathrm{a}=6-2 \\ \Rightarrow \mathrm{a}=4 \end{gathered}$$