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The population, [tex]p[/tex], of a town after [tex]t[/tex] years is represented using the equation [tex]p=10000(1.04)^{-t}[/tex]. Which of the following is an equivalent expression?

A. [tex]p=10000\left(\frac{1}{25}\right)^t[/tex]

B. [tex]p=10000\left(\frac{25}{26}\right)^t[/tex]

C. [tex]p=10000\left(\frac{26}{25}\right)^t[/tex]

D. [tex]p=10000\left(\frac{25}{1}\right)^t[/tex]

Answer :

We are given the expression
$$
p = 10000 (1.04)^{-t}.
$$
A negative exponent means we can write the expression as
$$
p = 10000 \left(\frac{1}{1.04}\right)^t.
$$

Next, we simplify the fraction $\frac{1}{1.04}$. Recognize that
$$
1.04 = \frac{26}{25},
$$
so
$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$

Substituting this back into the expression for $p$, we have
$$
p = 10000\left(\frac{25}{26}\right)^t.
$$

Thus, the equivalent expression is
$$
p = 10000\left(\frac{25}{26}\right)^t.
$$

This corresponds to option 2.

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