To determine who has more money in their account and the difference in their account balances, we need to simplify each expression for Sama and Jimena.
Simplifying Sama's expression:
Sama: \[ \frac{5^8 \cdot 5^4}{(5^3)^2} \]
First, simplify the numerator: \[ 5^8 \cdot 5^4 = 5^{8 + 4} = 5^{12} \]
Now simplify the denominator: \[ (5^3)^2 = 5^{3 \cdot 2} = 5^6 \]
Putting it together: \[ \frac{5^{12}}{5^6} = 5^{12 - 6} = 5^6 \]
Calculating \(5^6\): \[ 5^6 = 15625 \]
Simplifying Jimena's expression:
Jimena: \[ \frac{2^9 \cdot (2^{-2})^{-3}}{2} \]
First, simplify the exponent in the numerator: \[ (2^{-2})^{-3} = 2^{-2 \cdot -3} = 2^6 \]
Now simplify the numerator: \[ 2^9 \cdot 2^6 = 2^{9 + 6} = 2^{15} \]
Now, simplify the entire expression: \[ \frac{2^{15}}{2} = 2^{15 - 1} = 2^{14} \]
Calculating \(2^{14}\): \[ 2^{14} = 16384 \]
Comparing their amounts:
Sama has \(15625\) and Jimena has \(16384\).
Responses:
-
Who has more money in their account?
Jimena -
What is the difference in the amount of money that they have?
\[ 16384 - 15625 = 759 \]
Final Answers:
- Jimena
- 759