Let's simplify both expressions step by step to determine who has more money.
Sama's expression:
\[ Sama: \quad \frac{4^3 \cdot 4^{-4}}{(4^2)^{-5}} \]
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Simplify the numerator: \[ 4^3 \cdot 4^{-4} = 4^{3 - 4} = 4^{-1} = \frac{1}{4} \]
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Simplify the denominator: \[ (4^2)^{-5} = 4^{-10} \]
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Now, rewrite the entire expression: \[ \frac{4^{-1}}{4^{-10}} = 4^{-1 - (-10)} = 4^{-1 + 10} = 4^9 \]
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Calculate \(4^9\): \[ 4^9 = (2^2)^9 = 2^{18} = 262144 \]
So, Sama has \(262144\) dollars.
Jimena's expression:
\[ Jimena: \quad \frac{3 \cdot (3^8)^2}{(3^2)^3 \cdot 3^0} \]
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Simplify the numerator: \[ (3^8)^2 = 3^{16} \] So, the numerator becomes: \[ 3 \cdot 3^{16} = 3^{1 + 16} = 3^{17} \]
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Simplify the denominator: \[ (3^2)^3 = 3^{6} \quad \text{and} \quad 3^0 = 1 \] So, the denominator becomes: \[ 3^6 \cdot 1 = 3^6 \]
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Now, rewrite the entire expression: \[ \frac{3^{17}}{3^6} = 3^{17 - 6} = 3^{11} \]
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Calculate \(3^{11}\): \[ 3^{11} = 177147 \]
So, Jimena has \(177147\) dollars.
Comparison: Sama has \(262144\) dollars, while Jimena has \(177147\) dollars.
Thus, Sama has more money.