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Answer :
To solve the problem of finding an expression equivalent to [tex]\( p = 10000\left(\frac{1}{25}\right)^t \)[/tex], we need to understand the mathematical concept of equivalent expressions.
1. Understanding the Expression: The original expression is [tex]\( p = 10000\left(\frac{1}{25}\right)^t \)[/tex]. This tells us that the population [tex]\( p \)[/tex] is reduced by a factor of [tex]\(\frac{1}{25}\)[/tex] for every year [tex]\( t \)[/tex].
2. Finding Equivalent Expressions: An equivalent expression should have the same effect on the population over time. To find this, you seek an expression where raising something to the power of [tex]\( t \)[/tex] results in the same outcome.
3. Exponent Rules: Remember that expressions like [tex]\( \left(\frac{a}{b}\right)^t = \left(\frac{b}{a}\right)^{-t} \)[/tex]. This is because raising a fraction to a positive power results in the same product as raising its reciprocal to a negative power.
4. Applying to Our Problem: We have [tex]\( \left(\frac{1}{25}\right)^t \)[/tex]. This can be expressed equivalently as [tex]\( \left(\frac{25}{1}\right)^{-t} = 25^{-t} \)[/tex].
5. Matching the Options: Among our choices:
- [tex]\( p = 10000\left(\frac{25}{1}\right)^t \)[/tex] is option 4. While it involves the reciprocal base, to truly match [tex]\( \left(\frac{1}{25}\right)^t \)[/tex], it would need a negative exponent to reverse the effect, which matches our earlier transformation to [tex]\( 25^{-t} \)[/tex].
Consequently, the equivalent expression that maintains the same population dynamics over time is indeed similar to the conceptual understanding in option 4. Therefore, the choice that matches the criteria for equivalency is:
[tex]\[ p = 10000\left(\frac{25}{1}\right)^{-t} \][/tex]
This means option 4 would be the correct equivalent expression. However, note the structure needs that negative exponent which correctly reflects the original mathematical dynamics.
1. Understanding the Expression: The original expression is [tex]\( p = 10000\left(\frac{1}{25}\right)^t \)[/tex]. This tells us that the population [tex]\( p \)[/tex] is reduced by a factor of [tex]\(\frac{1}{25}\)[/tex] for every year [tex]\( t \)[/tex].
2. Finding Equivalent Expressions: An equivalent expression should have the same effect on the population over time. To find this, you seek an expression where raising something to the power of [tex]\( t \)[/tex] results in the same outcome.
3. Exponent Rules: Remember that expressions like [tex]\( \left(\frac{a}{b}\right)^t = \left(\frac{b}{a}\right)^{-t} \)[/tex]. This is because raising a fraction to a positive power results in the same product as raising its reciprocal to a negative power.
4. Applying to Our Problem: We have [tex]\( \left(\frac{1}{25}\right)^t \)[/tex]. This can be expressed equivalently as [tex]\( \left(\frac{25}{1}\right)^{-t} = 25^{-t} \)[/tex].
5. Matching the Options: Among our choices:
- [tex]\( p = 10000\left(\frac{25}{1}\right)^t \)[/tex] is option 4. While it involves the reciprocal base, to truly match [tex]\( \left(\frac{1}{25}\right)^t \)[/tex], it would need a negative exponent to reverse the effect, which matches our earlier transformation to [tex]\( 25^{-t} \)[/tex].
Consequently, the equivalent expression that maintains the same population dynamics over time is indeed similar to the conceptual understanding in option 4. Therefore, the choice that matches the criteria for equivalency is:
[tex]\[ p = 10000\left(\frac{25}{1}\right)^{-t} \][/tex]
This means option 4 would be the correct equivalent expression. However, note the structure needs that negative exponent which correctly reflects the original mathematical dynamics.
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