To compare Sama's and Jimena's bank accounts, we need to simplify both expressions.
Sama's Account: \[ 4^3 \cdot 4^{-4} / (4^2)^{-5} \]
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Simplify the numerator: \[ 4^3 \cdot 4^{-4} = 4^{3 - 4} = 4^{-1} \]
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Simplify the denominator: \[ (4^2)^{-5} = 4^{-10} \]
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Now substitute back into the equation: \[ \frac{4^{-1}}{4^{-10}} = 4^{-1 - (-10)} = 4^{-1 + 10} = 4^9 \]
So, Sama has \( 4^9 \).
Jimena's Account: \[ 3 \cdot (3^8)^2 / (3^2)^3 \cdot 3^0 \]
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Simplify the expression: \[ (3^8)^2 = 3^{16} \] \[ (3^2)^3 = 3^6 \]
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Now substitute back: \[ 3 \cdot 3^{16} / 3^6 \cdot 3^0 = 3^1 \cdot 3^{16} / 3^6 = 3^{1 + 16 - 6} = 3^{11} \]
So, Jimena has \( 3^{11} \).
Now we have:
- Sama: \( 4^9 \)
- Jimena: \( 3^{11} \)
To compare \( 4^9 \) and \( 3^{11} \), we can convert them into a common base or use logarithms.
Calculating the values:
- \( 4^9 = (2^2)^9 = 2^{18} \)
- \( 3^{11} \) remains \( 3^{11} \)
Now we need to compare \( 2^{18} \) and \( 3^{11} \).
Using logarithms for a more precise comparison:
- Convert into the same logarithmic base, for example, base 10:
- \( 2^{18} \) can be approximated using: \[ 18 \log_{10}(2) \approx 18 \cdot 0.301 \approx 5.418 \]
- \( 3^{11} \) can be approximated using: \[ 11 \log_{10}(3) \approx 11 \cdot 0.477 \approx 5.247 \]
Thus, \( 5.418 > 5.247 \) which implies \( 4^9 > 3^{11} \).
Therefore, Sama has more money in her account.
The final answer is: Sama.