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Answer :
To solve for an equivalent expression to the given equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], let's break it down step by step:
1. Understand the Original Expression:
The original expression for the population is [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Convert the Base to a Fraction:
We start by converting the base [tex]\( 1.04 \)[/tex] to a fraction. We can express [tex]\( 1.04 \)[/tex] as [tex]\( \frac{104}{100} \)[/tex], which simplifies to [tex]\( \frac{26}{25} \)[/tex].
3. Apply the Negative Exponent:
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten using the fraction:
[tex]\[
\left(\frac{26}{25}\right)^{-t}
\][/tex]
4. Simplify the Exponent:
Applying the rule for negative exponents, [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\)[/tex], we rewrite:
[tex]\[
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}
\][/tex]
5. Find the Equivalent Expression:
Now, substitute back into the original equation:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^{t}
\][/tex]
Thus, the equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^{t} \][/tex]
This means the expression given by [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original expression.
1. Understand the Original Expression:
The original expression for the population is [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Convert the Base to a Fraction:
We start by converting the base [tex]\( 1.04 \)[/tex] to a fraction. We can express [tex]\( 1.04 \)[/tex] as [tex]\( \frac{104}{100} \)[/tex], which simplifies to [tex]\( \frac{26}{25} \)[/tex].
3. Apply the Negative Exponent:
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten using the fraction:
[tex]\[
\left(\frac{26}{25}\right)^{-t}
\][/tex]
4. Simplify the Exponent:
Applying the rule for negative exponents, [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\)[/tex], we rewrite:
[tex]\[
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}
\][/tex]
5. Find the Equivalent Expression:
Now, substitute back into the original equation:
[tex]\[
p = 10000 \left(\frac{25}{26}\right)^{t}
\][/tex]
Thus, the equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^{t} \][/tex]
This means the expression given by [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original expression.
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