Question

Two students have bank accounts:
Sama: $ \frac{4^3
ewline "." 4^{-4}}{(4^2)^{-5}}
Jimena: $ \frac{3
ewline "." (3^8)^2}{(3^2)^3
ewline "." 3^0}
Which person has the most money in their account?
(1 point)
Sama
You cannot compare the total amount because the bases are different.
Jimena
They have the same amount of money

Answer 1

Let's simplify the expressions for Sama and Jimena.

For Sama:
\[
Sama = \frac{4^3 \cdot 4^{-4}}{(4^2)^{-5}}
\]

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

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

Then the expression for Sama becomes:
\[
Sama = \frac{4^{-1}}{4^{-10}} = 4^{-1 - (-10)} = 4^{9}
\]

Now calculate \(4^9\):
\[
4^9 = (2^2)^9 = 2^{18}
\]

For Jimena:
\[
Jimena = \frac{3 \cdot (3^8)^2}{(3^2)^3 \cdot 3^0}
\]

First, simplify the numerator:
\[
(3^8)^2 = 3^{16} \implies 3 \cdot 3^{16} = 3^{1 + 16} = 3^{17}
\]

Now simplify the denominator:
\[
(3^2)^3 = 3^{6} \text{ and } 3^0 = 1 \implies 3^6 \cdot 3^0 = 3^{6}
\]

Then the expression for Jimena becomes:
\[
Jimena = \frac{3^{17}}{3^{6}} = 3^{17 - 6} = 3^{11}
\]

Now, we need to compare \(4^9\) and \(3^{11}\).

Convert both to powers of 2, if possible:
\[
4^9 = (2^2)^9 = 2^{18}
\]
\[
3^{11} = 3^{11} \text{ (cannot be expressed easily in base 2)}
\]

Now, calculate approximate values:
\[
2^{18} = 262144
\]
\[
3^{11} = 177147
\]

So if we compare \(262144\) (for Sama) to \(177147\) (for Jimena), we find that:
\[
262144 > 177147
\]

Thus, Sama has the most money in her account.

The answer is:
**Sama**.