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Answer :
Sure! Let's break down the problem to find an equivalent expression for [tex]\( p = 10000(1.04)^{-t} \)[/tex].
1. Understand the problem: We have the original expression that represents the population after [tex]\( t \)[/tex] years: [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Simplify the expression: The term [tex]\( (1.04)^{-t} \)[/tex] can be rewritten using the property of exponents that says [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. So, we can express [tex]\( (1.04)^{-t} \)[/tex] as [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex].
3. Calculate the fraction: Let's simplify the fraction [tex]\( \frac{1}{1.04} \)[/tex]:
[tex]\[
\frac{1}{1.04} = \frac{25}{26}
\][/tex]
This simplification means [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex] is the same as [tex]\( \left(\frac{25}{26}\right)^t \)[/tex].
4. Write the equivalent expression: Substitute this back into the population equation. We have:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
By following these steps, we find that the equivalent expression is:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
So, the correct choice is:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
1. Understand the problem: We have the original expression that represents the population after [tex]\( t \)[/tex] years: [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Simplify the expression: The term [tex]\( (1.04)^{-t} \)[/tex] can be rewritten using the property of exponents that says [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. So, we can express [tex]\( (1.04)^{-t} \)[/tex] as [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex].
3. Calculate the fraction: Let's simplify the fraction [tex]\( \frac{1}{1.04} \)[/tex]:
[tex]\[
\frac{1}{1.04} = \frac{25}{26}
\][/tex]
This simplification means [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex] is the same as [tex]\( \left(\frac{25}{26}\right)^t \)[/tex].
4. Write the equivalent expression: Substitute this back into the population equation. We have:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
By following these steps, we find that the equivalent expression is:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
So, the correct choice is:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^t
\][/tex]
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