Question

Two students have bank accounts:
Sama: 4 to the power of three times 4 to the power of negative four divided by (4 to the power of two) to the power of negative five
Jimena: 3 times (3 to the power of eight) to the power of two divided by (3 to the power of two) to the power of three times 3 to the power zero

Which person has the most money in their account

Answer 1

Let's simplify each student's bank account expression step by step.

**Sama's expression:**
\[
\text{Sama's amount} = \frac{4^3 \times 4^{-4}}{(4^2)^{-5}}
\]

1. Simplifying the numerator:
\[
4^3 \times 4^{-4} = 4^{3 + (-4)} = 4^{-1} = \frac{1}{4}
\]

2. Simplifying the denominator:
\[
(4^2)^{-5} = 4^{-10}
\]

So, we rewrite the expression:
\[
\text{Sama's amount} = \frac{4^{-1}}{4^{-10}} = 4^{-1 - (-10)} = 4^{9}
\]

Calculating \( 4^9 \):
\[
4^9 = (2^2)^9 = 2^{18} = 262144
\]

**Jimena's expression:**
\[
\text{Jimena's amount} = \frac{3 \times (3^8)^2}{(3^2)^3 \times 3^0}
\]

1. Simplifying the numerator:
\[
(3^8)^2 = 3^{8 \times 2} = 3^{16}
\]
Thus,
\[
\text{Numerator} = 3 \times 3^{16} = 3^{1 + 16} = 3^{17}
\]

2. Simplifying the denominator:
\[
(3^2)^3 = 3^{2 \times 3} = 3^6
\]
Also, \( 3^0 = 1 \), so the denominator becomes:
\[
\text{Denominator} = 3^6 \times 1 = 3^6
\]

Thus, we rewrite Jimena's amount as:
\[
\text{Jimena's amount} = \frac{3^{17}}{3^6} = 3^{17 - 6} = 3^{11}
\]

Calculating \( 3^{11} \):
\[
3^{11} = 177147
\]

**Final Comparison:**
Now we compare:
- Sama's amount: \( 4^9 = 262144 \)
- Jimena's amount: \( 3^{11} = 177147 \)

Since \( 262144 > 177147 \), we conclude:

**Sama has more money in her account.**