Discussion
If 12500 = \(5^{x} \times 2^{y}\), then which of the following can be expressed in the form \(5^{x-1} \times 2^{2y}\) ?
Answer: B
The number is 10000.
Write 12500 in prime factors:
\[ 12500 = 125 \times 100 = 5^{5} \times 2^{2} \]
So \(x = 5\) and \(y = 2\). Then
\[ \begin{aligned} 5^{x - 1} \times 2^{2y} &= 5^{4} \times 2^{4} \\ &= 625 \times 16 = 10000 \end{aligned} \]
Equivalently, \(5^{4} \times 2^{4} = 10^{4}\). 2500 is \(5^{4} \times 2^{2}\), which forgets to double the power of 2, and 62500 is \(5^{6} \times 2^{2}\).
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