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Answer :
Sure, let's work through the problem step-by-step to find the equivalent expression for the given equation.
The original expression for the population of the town is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find which of the given expressions is equivalent to this.
First, let's express [tex]\(1.04\)[/tex] as a fraction. We know that:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
So, we can rewrite the original expression using this fraction:
[tex]\[ p = 10000\left(\frac{26}{25}\right)^{-t} \][/tex]
When you have a negative exponent, it means you take the reciprocal of the base. Therefore:
[tex]\[ \left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^t \][/tex]
Now substitute this back into the expression for [tex]\(p\)[/tex]:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This matches one of the given options, which is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
So, the equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
The original expression for the population of the town is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find which of the given expressions is equivalent to this.
First, let's express [tex]\(1.04\)[/tex] as a fraction. We know that:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
So, we can rewrite the original expression using this fraction:
[tex]\[ p = 10000\left(\frac{26}{25}\right)^{-t} \][/tex]
When you have a negative exponent, it means you take the reciprocal of the base. Therefore:
[tex]\[ \left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^t \][/tex]
Now substitute this back into the expression for [tex]\(p\)[/tex]:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This matches one of the given options, which is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
So, the equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
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