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1. (25)^7.5 × (5)^2.5 ÷ (125)^1.5 = 5^?

A. 8.5

B. 13

C. 16

D. 17.5

Solution:

\begin{aligned} Let {(25)}^{7.5} \times {(5)}^{2.5} \div {(125)}^{1.5} = 5^{x} \\ Then, \frac{{(5^{2})}^{7.5} \times {(5)}^{2.5}}{{(5^{3})}^{1.5}} = 5^{x} \\ \Rightarrow \frac{5^{(2 \times 7.5)} \times 5^{2.5}}{5^{(3 \times 1.5)}} = 5^{x} \\ \Rightarrow \frac{5^{15} \times 5^{2.5}}{5^{4.5}} = 5^{x} \\ \Rightarrow 5^{x} = 5^{(15 + 2.5 - 4.5)} \\ \Rightarrow 5^{x} = 5^{13} \\ ∴ x = 13 \end{aligned}

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