Sure, let's tackle the expression step-by-step.
We start with the given expression:
[tex]\[
a \left(2^3 \times 18\right)^4 : \left(3 \times 16^2\right)^2
\][/tex]
First, we'll simplify the inner parts of our expression before raising them to their respective powers.
### Step 1: Simplify inside the parentheses
1. Calculate [tex]\(2^3\)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
2. Multiply [tex]\(8\)[/tex] by [tex]\(18\)[/tex]:
[tex]\[ 8 \times 18 = 144 \][/tex]
Now we have:
[tex]\[
a \left(144\right)^4 : \left(3 \times 16^2\right)^2
\][/tex]
Next, let's handle the second part inside the parentheses:
3. Calculate [tex]\(16^2\)[/tex]:
[tex]\[ 16^2 = 256 \][/tex]
4. Multiply [tex]\(256\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[ 3 \times 256 = 768 \][/tex]
Which means we have:
[tex]\[
a \left(144\right)^4 : \left(768\right)^2
\][/tex]
### Step 2: Raise the simplified terms to their respective exponents
Next, we raise each term to the power specified.
1. Calculate [tex]\((144)^4\)[/tex]:
[tex]\[ 144^4 = 429981696 \][/tex]
2. Calculate [tex]\((768)^2\)[/tex]:
[tex]\[ 768^2 = 589824 \][/tex]
This transforms our expression into:
[tex]\[
a \frac{429981696}{589824}
\][/tex]
### Step 3: Divide the results
Now, perform the division:
[tex]\[ \frac{429981696}{589824} = 729 \][/tex]
### Conclusion
Therefore, the simplified expression is:
[tex]\[
729a
\][/tex]
So we have shown step-by-step that:
[tex]\[
a \left(2^3 \times 18\right)^4 : \left(3 \times 16^2\right)^2 = 729a
\][/tex]
This means if we evaluate this expression (ignoring the multiplication by [tex]\(a\)[/tex]) the result is [tex]\(729\)[/tex].
Thus, the final answer is:
[tex]\[
729
\][/tex]
