Sure! Let's simplify the given expression step-by-step:
The expression we need to simplify is:
[tex]\[ 4^3 \times 3^3 + \frac{12^3}{12^4} \][/tex]
### Step 1: Simplify the first term
First, we calculate [tex]\( 4^3 \times 3^3 \)[/tex].
[tex]\[ 4^3 = 64 \][/tex]
[tex]\[ 3^3 = 27 \][/tex]
[tex]\[ 4^3 \times 3^3 = 64 \times 27 = 1728 \][/tex]
So, the first term simplifies to:
[tex]\[ 1728 \][/tex]
### Step 2: Simplify the second term
Next, we simplify [tex]\(\frac{12^3}{12^4}\)[/tex].
When dividing like bases, we subtract the exponents:
[tex]\[ \frac{12^3}{12^4} = 12^{3-4} = 12^{-1} \][/tex]
[tex]\[ 12^{-1} = \frac{1}{12} \][/tex]
Thus, the second term simplifies to:
[tex]\[ \frac{1}{12} \approx 0.083333\][/tex]
### Step 3: Combine the simplified results
Now, we add the simplified terms together:
[tex]\[ 1728 + 12^{-1} \][/tex]
### Step 4: Final result
The final expression is:
[tex]\[ 1728 + \frac{1}{12} \][/tex]
Comparing this with the given choices:
[tex]\[
(a) \ 12^{3-1} \\
(b) \ 12^6 + 12^{-1} \\
(c) \ 12^{9-1} \\
(d) \ 12^3 + 12^{-1}
\][/tex]
It is clear that the answer matches choice [tex]\( \boxed{d} \)[/tex] which is:
[tex]\[ 12^3 + 12^{-1} \][/tex]
