Question

Two students have bank accounts:

Sama: $ 4 superscript3 ⋅ 4 − superscript4 / (4 superscript2) − superscript5


Jimena: $ 3 ⋅ (3 superscript8) superscript2 (3 superscript2) superscript3 ⋅ 3 superscript0



Which person has the most money in their account?

(1 point)
Responses

They have the same amount of money
They have the same amount of money

Sama
Sama

You cannot compare the total amount because the bases are different.
You cannot compare the total amount because the bases are different.

Jimena

Answer 1

To determine who has the most money in their account, let's simplify the expressions for each student's bank account.

**Sama's Account:**

Sama's expression is:
\[
\frac{4^{3} \cdot 4^{-4}}{4^{2} \cdot 4^{-5}}
\]

First, let's simplify the numerator:
\[
4^{3} \cdot 4^{-4} = 4^{3-4} = 4^{-1}
\]

Now for the denominator:
\[
4^{2} \cdot 4^{-5} = 4^{2-5} = 4^{-3}
\]

Now we substitute back into the equation:
\[
\frac{4^{-1}}{4^{-3}} = 4^{-1 - (-3)} = 4^{-1 + 3} = 4^{2} = 16
\]

So, Sama has $16.

---

**Jimena's Account:**

Jimena's expression is:
\[
3 \cdot (3^{8})^{2} \cdot (3^{2})^{3} \cdot 3^{0}
\]

Now let's simplify each part:
\[
(3^{8})^{2} = 3^{8 \cdot 2} = 3^{16}
\]
\[
(3^{2})^{3} = 3^{2 \cdot 3} = 3^{6}
\]
\[
3^{0} = 1
\]

Now we can combine everything:
\[
3 \cdot 3^{16} \cdot 3^{6} \cdot 1 = 3^{1} \cdot 3^{16} \cdot 3^{6} = 3^{1 + 16 + 6} = 3^{23}
\]

Now, we can calculate \(3^{23}\) but it's a very large number. We can use a calculator or leave it as is.

---

**Comparison:**

- Sama has $16.
- Jimena has $3^{23}, which is a very large number (exact value is 94143178827).

Clearly, \(3^{23} \gg 16\). Thus, Jimena has more money.

**Final Answer: Jimena**