Question

Two students have bank accounts:

Sama: $ 5^8 ⋅ 5^4/(5^3)^2


Jimena: $ 2^9 ⋅ (2^−2)^−3/2


(2 points)
Put responses in the correct input to answer the question.
Who has more money in their account? Response area

What is the difference in the amount of money that they have? Response area

Answer 1

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:

1. **Who has more money in their account?**
Jimena

2. **What is the difference in the amount of money that they have?**
\[
16384 - 15625 = 759
\]

### Final Answers:

1. Jimena
2. 759