Let's simplify the given expression step by step and then compare it with the choices provided.
The given expression is:
[tex]\[
\frac{6^3 \cdot 2^5}{2^3}
\][/tex]
### Step-by-Step Simplification
1. Simplify [tex]\(6^3\)[/tex]:
Notice that [tex]\(6 = 2 \cdot 3\)[/tex]. So,
[tex]\[
6^3 = (2 \cdot 3)^3 = 2^3 \cdot 3^3
\][/tex]
2. Substitute [tex]\(6^3\)[/tex] back into the expression:
[tex]\[
\frac{6^3 \cdot 2^5}{2^3} = \frac{(2^3 \cdot 3^3) \cdot 2^5}{2^3}
\][/tex]
3. Combine the powers of 2:
Since we have [tex]\(2^3 \cdot 2^5\)[/tex] in the numerator,
[tex]\[
2^3 \cdot 2^5 = 2^{3+5} = 2^8
\][/tex]
So now the expression becomes:
[tex]\[
\frac{2^8 \cdot 3^3}{2^3}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{2^8 \cdot 3^3}{2^3} = 2^{8-3} \cdot 3^3 = 2^5 \cdot 3^3
\][/tex]
Now, the simplified form of the expression is:
[tex]\[
2^5 \cdot 3^3
\][/tex]
### Comparing with Provided Choices
Let's check if any of the provided choices match [tex]\(2^5 \cdot 3^3\)[/tex]:
1. Choice: [tex]\(2^3 \cdot 3^3\)[/tex]:
[tex]\[
2^3 \cdot 3^3 = 8 \cdot 27 = 216
\][/tex]
This is not equal to [tex]\(2^5 \cdot 3^3\)[/tex].
2. Choice: [tex]\(12^3\)[/tex]:
[tex]\[
12^3 = 1728
\][/tex]
This is not equal to [tex]\(2^5 \cdot 3^3\)[/tex].
3. Choice: [tex]\(6^3\)[/tex]:
[tex]\[
6^3 = 216
\][/tex]
This is not equal to [tex]\(2^5 \cdot 3^3\)[/tex].
4. Choice: [tex]\(12^6\)[/tex]:
[tex]\[
12^6 = 2985984
\][/tex]
This is not equal to [tex]\(2^5 \cdot 3^3\)[/tex].
5. Choice: [tex]\(2^6 \cdot 3^3\)[/tex]:
[tex]\[
2^6 \cdot 3^3 = 64 \cdot 27 = 1728
\][/tex]
This is not equal to [tex]\(2^5 \cdot 3^3\)[/tex].
After evaluating all the choices, none of the given expressions match the simplified form [tex]\(2^5 \cdot 3^3\)[/tex]. Hence, none of the provided choices are equal to [tex]\(\frac{6^3 \cdot 2^5}{2^3}\)[/tex].
