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Answer :
We start with the population equation:
[tex]$$
p = 10000\,(1.04)^{-t}.
$$[/tex]
Using the exponent property, a negative exponent can be rewritten as a reciprocal:
[tex]$$
p = 10000\left(\frac{1}{1.04}\right)^t.
$$[/tex]
Next, we express [tex]$\frac{1}{1.04}$[/tex] as a common fraction. Notice that:
[tex]$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}},
$$[/tex]
because [tex]$1.04 = \frac{26}{25}$[/tex]. Taking the reciprocal gives:
[tex]$$
\frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$[/tex]
Substituting back into our equation, we have:
[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]
Comparing this with the multiple-choice options, the equivalent expression is:
[tex]$$
10000\left(\frac{25}{26}\right)^t.
$$[/tex]
Thus, the correct answer is option 2.
[tex]$$
p = 10000\,(1.04)^{-t}.
$$[/tex]
Using the exponent property, a negative exponent can be rewritten as a reciprocal:
[tex]$$
p = 10000\left(\frac{1}{1.04}\right)^t.
$$[/tex]
Next, we express [tex]$\frac{1}{1.04}$[/tex] as a common fraction. Notice that:
[tex]$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}},
$$[/tex]
because [tex]$1.04 = \frac{26}{25}$[/tex]. Taking the reciprocal gives:
[tex]$$
\frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$[/tex]
Substituting back into our equation, we have:
[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]
Comparing this with the multiple-choice options, the equivalent expression is:
[tex]$$
10000\left(\frac{25}{26}\right)^t.
$$[/tex]
Thus, the correct answer is option 2.
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