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Answer :
Sure, let's solve this problem step-by-step to find the equivalent expression for the population equation given.
The original equation for the population, [tex]\( p \)[/tex], is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We want to find an equivalent expression by rewriting the term [tex]\( (1.04)^{-t} \)[/tex] in a different form. Let's break it down:
1. Understanding the Exponent:
- The negative exponent [tex]\((-t)\)[/tex] signifies that we are dealing with the reciprocal. So, [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \frac{1}{(1.04)^t} \)[/tex].
2. Rewriting the Base:
- The number 1.04 can be rewritten as a fraction. We recognize that:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
- Therefore, [tex]\( (1.04)^{-t} \)[/tex] can also be rewritten as [tex]\( (\frac{26}{25})^{-t} \)[/tex].
3. Using the Reciprocal Rule:
- The reciprocal of a fraction raised to a power means we flip the fraction and change the sign of the exponent:
[tex]\[ (\frac{26}{25})^{-t} = (\frac{25}{26})^t \][/tex]
4. Substituting Back into the Original Equation:
- Substitute the equivalent expression back into the original equation:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Now we have successfully rewritten the original expression to:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This matches the option:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Hence, the equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
The original equation for the population, [tex]\( p \)[/tex], is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We want to find an equivalent expression by rewriting the term [tex]\( (1.04)^{-t} \)[/tex] in a different form. Let's break it down:
1. Understanding the Exponent:
- The negative exponent [tex]\((-t)\)[/tex] signifies that we are dealing with the reciprocal. So, [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \frac{1}{(1.04)^t} \)[/tex].
2. Rewriting the Base:
- The number 1.04 can be rewritten as a fraction. We recognize that:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
- Therefore, [tex]\( (1.04)^{-t} \)[/tex] can also be rewritten as [tex]\( (\frac{26}{25})^{-t} \)[/tex].
3. Using the Reciprocal Rule:
- The reciprocal of a fraction raised to a power means we flip the fraction and change the sign of the exponent:
[tex]\[ (\frac{26}{25})^{-t} = (\frac{25}{26})^t \][/tex]
4. Substituting Back into the Original Equation:
- Substitute the equivalent expression back into the original equation:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Now we have successfully rewritten the original expression to:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This matches the option:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Hence, the equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
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