Question

Sama: $ 4^3 x 4^-4 / (4^2)^-5

Jimena: $ 3 x (3^8)^2 / (3^2)^2 x 3^0

Who has more money?

Answer 1

Let's simplify the expressions for Sama and Jimena step by step.

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

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

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

3. Now substituting back into the expression:
\[
\frac{4^{-1}}{4^{-10}} = 4^{-1 - (-10)} = 4^{-1 + 10} = 4^9
\]

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

Now, Sama has \(262144\).

---

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

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

2. Simplifying the denominator:
\[
(3^2)^2 = 3^4
\]
And since \(3^0 = 1\),
\[
3^4 \times 3^0 = 3^4
\]

3. Now substituting back into the expression:
\[
\frac{3^{17}}{3^4} = 3^{17 - 4} = 3^{13}
\]

4. Calculating \(3^{13}\):
To compute \(3^{13}\):
\[
3^{13} = 1594323
\]

Now, Jimena has \(1594323\).

---

**Comparison:**
- Sama has \(262144\).
- Jimena has \(1594323\).

Clearly, Jimena has more money.