Let's calculate the amounts for both Sama and Jimena.
For Sama:
The expression is:
\[ \frac{5^8 \cdot 5^4}{(5^3)^2} \]
First, simplify the numerator using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
\[ 5^8 \cdot 5^4 = 5^{8+4} = 5^{12} \]
Now simplify the denominator:
\[ (5^3)^2 = 5^{3 \cdot 2} = 5^6 \]
Now, divide the numerator by the denominator:
\[ \frac{5^{12}}{5^6} = 5^{12-6} = 5^6 \]
For Jimena:
The expression is:
\[ \frac{2^9 \cdot (2^{-2})^{-3}}{2} \]
First, simplify \((2^{-2})^{-3}\) using the property \( (a^m)^n = a^{m \cdot n} \):
\[ (2^{-2})^{-3} = 2^{-2 \cdot -3} = 2^6 \]
Now, multiply this result with \(2^9\):
\[ 2^9 \cdot 2^6 = 2^{9+6} = 2^{15} \]
Now, divide by \(2\):
\[ \frac{2^{15}}{2^1} = 2^{15-1} = 2^{14} \]
Final calculations:
Now we need to compare \(5^6\) and \(2^{14}\).
- Calculate \(5^6\):
\[ 5^6 = 15625 \]
- Calculate \(2^{14}\):
\[ 2^{14} = 16384 \]
Comparison: Since \(16384 > 15625\),
-
Who has more money in their account? Jimena
-
What is the difference in the amount of money that they have?
To find the difference:
\[ 2^{14} - 5^6 = 16384 - 15625 = 759 \]
Final responses:
- Who has more money in their account? Jimena
- What is the difference in the amount of money that they have? 759
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