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Answer :
To solve this problem step-by-step, let's look at the given equation and the options we're comparing it to.
The given equation is:
[tex]\[ p = 10000 (1.04)^{-t} \][/tex]
We want to find which of the following expressions is equivalent to this equation:
1. [tex]\( p = 10000 \left(\frac{1}{25}\right)^t \)[/tex]
2. [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]
3. [tex]\( p = 10000 \left(\frac{26}{25}\right)^t \)[/tex]
Let's analyze the given equation and use properties of exponents to find the equivalent expression.
### Step-by-Step Solution:
1. Understanding the given equation:
[tex]\[ p = 10000 (1.04)^{-t} \][/tex]
2. Rewrite the base [tex]\( 1.04 \)[/tex] in a different form:
Recall that [tex]\( 1.04 = \frac{104}{100} = \frac{26}{25} \)[/tex].
3. Substitute [tex]\( \frac{26}{25} \)[/tex] into the given equation:
[tex]\[ p = 10000 \left( \frac{26}{25} \right)^{-t} \][/tex]
4. Apply the property of exponents:
Using the property of exponents [tex]\( a^{-b} = \left( \frac{1}{a} \right)^b \)[/tex], we can rewrite [tex]\( \left( \frac{26}{25} \right)^{-t} \)[/tex] as:
[tex]\[ \left( \frac{26}{25} \right)^{-t} = \left( \frac{25}{26} \right)^t \][/tex]
5. Substitute this result back into the equation:
[tex]\[ p = 10000 \left( \frac{25}{26} \right)^t \][/tex]
So the expression equivalent to the given equation [tex]\( p = 10000 (1.04)^{-t} \)[/tex] is:
[tex]\[ p = 10000 \left( \frac{25}{26} \right)^t \][/tex]
### Conclusion
The correct equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Thus, the answer is:
[tex]\[ \boxed{10000 \left( \frac{25}{26} \right)^t} \][/tex]
The given equation is:
[tex]\[ p = 10000 (1.04)^{-t} \][/tex]
We want to find which of the following expressions is equivalent to this equation:
1. [tex]\( p = 10000 \left(\frac{1}{25}\right)^t \)[/tex]
2. [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]
3. [tex]\( p = 10000 \left(\frac{26}{25}\right)^t \)[/tex]
Let's analyze the given equation and use properties of exponents to find the equivalent expression.
### Step-by-Step Solution:
1. Understanding the given equation:
[tex]\[ p = 10000 (1.04)^{-t} \][/tex]
2. Rewrite the base [tex]\( 1.04 \)[/tex] in a different form:
Recall that [tex]\( 1.04 = \frac{104}{100} = \frac{26}{25} \)[/tex].
3. Substitute [tex]\( \frac{26}{25} \)[/tex] into the given equation:
[tex]\[ p = 10000 \left( \frac{26}{25} \right)^{-t} \][/tex]
4. Apply the property of exponents:
Using the property of exponents [tex]\( a^{-b} = \left( \frac{1}{a} \right)^b \)[/tex], we can rewrite [tex]\( \left( \frac{26}{25} \right)^{-t} \)[/tex] as:
[tex]\[ \left( \frac{26}{25} \right)^{-t} = \left( \frac{25}{26} \right)^t \][/tex]
5. Substitute this result back into the equation:
[tex]\[ p = 10000 \left( \frac{25}{26} \right)^t \][/tex]
So the expression equivalent to the given equation [tex]\( p = 10000 (1.04)^{-t} \)[/tex] is:
[tex]\[ p = 10000 \left( \frac{25}{26} \right)^t \][/tex]
### Conclusion
The correct equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
Thus, the answer is:
[tex]\[ \boxed{10000 \left( \frac{25}{26} \right)^t} \][/tex]
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