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Answer :
We start with the given equation:
[tex]$$
p = 10000(1.04)^{-t}.
$$[/tex]
Step 1. Express 1.04 as a fraction
Notice that
[tex]$$
1.04 = \frac{26}{25}.
$$[/tex]
Step 2. Substitute the fraction into the equation
Replacing [tex]$1.04$[/tex] with [tex]$\frac{26}{25}$[/tex] yields
[tex]$$
p = 10000\left(\frac{26}{25}\right)^{-t}.
$$[/tex]
Step 3. Apply the negative exponent rule
Recall that for any nonzero numbers [tex]$a$[/tex] and [tex]$b$[/tex] and exponent [tex]$t$[/tex]:
[tex]$$
\left(\frac{a}{b}\right)^{-t} = \left(\frac{b}{a}\right)^{t}.
$$[/tex]
Thus,
[tex]$$
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}.
$$[/tex]
Step 4. Write the equivalent expression
The population equation can now be written as:
[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]
This expression is equivalent to the original equation. Therefore, the correct equivalent expression is:
[tex]$$
\boxed{p = 10000\left(\frac{25}{26}\right)^t}.
$$[/tex]
[tex]$$
p = 10000(1.04)^{-t}.
$$[/tex]
Step 1. Express 1.04 as a fraction
Notice that
[tex]$$
1.04 = \frac{26}{25}.
$$[/tex]
Step 2. Substitute the fraction into the equation
Replacing [tex]$1.04$[/tex] with [tex]$\frac{26}{25}$[/tex] yields
[tex]$$
p = 10000\left(\frac{26}{25}\right)^{-t}.
$$[/tex]
Step 3. Apply the negative exponent rule
Recall that for any nonzero numbers [tex]$a$[/tex] and [tex]$b$[/tex] and exponent [tex]$t$[/tex]:
[tex]$$
\left(\frac{a}{b}\right)^{-t} = \left(\frac{b}{a}\right)^{t}.
$$[/tex]
Thus,
[tex]$$
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}.
$$[/tex]
Step 4. Write the equivalent expression
The population equation can now be written as:
[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]
This expression is equivalent to the original equation. Therefore, the correct equivalent expression is:
[tex]$$
\boxed{p = 10000\left(\frac{25}{26}\right)^t}.
$$[/tex]
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