To simplify the given expression [tex]\(\left(\frac{a c^4}{3 b^3}\right)^2\)[/tex], let's work through it step-by-step.
### Step 1: Understanding the expression
We start with:
[tex]\[ \left(\frac{a c^4}{3 b^3}\right)^2 \][/tex]
### Step 2: Apply the exponent to the numerator and the denominator
When we square a fraction, we square both the numerator and the denominator separately:
[tex]\[ \left(\frac{a c^4}{3 b^3}\right)^2 = \frac{(a c^4)^2}{(3 b^3)^2} \][/tex]
### Step 3: Simplify the numerator
Next, we square each factor inside the parenthesis in the numerator:
[tex]\[ (a c^4)^2 = a^2 (c^4)^2 = a^2 c^{4 \cdot 2} = a^2 c^8 \][/tex]
### Step 4: Simplify the denominator
Similarly, we square each factor inside the parenthesis in the denominator:
[tex]\[ (3 b^3)^2 = 3^2 (b^3)^2 = 9 b^{3 \cdot 2} = 9 b^6 \][/tex]
### Step 5: Combine the results
Now that we have simplified the numerator and the denominator, we can write down the final expression:
[tex]\[ \frac{(a c^4)^2}{(3 b^3)^2} = \frac{a^2 c^8}{9 b^6} \][/tex]
Therefore, the simplified form of the expression [tex]\(\left(\frac{a c^4}{3 b^3}\right)^2\)[/tex] is:
[tex]\[ \boxed{\frac{a^2 c^8}{9 b^6}} \][/tex]
None of the provided answer choices seems to directly match the correct simplified form [tex]\(\frac{a^2 c^8}{9 b^6}\)[/tex]. If we consider typographical corrections, perhaps there is a mistake in one of the choices. But based on correct simplification, the answer is indeed:
[tex]\[ \frac{a^2 c^8}{9 b^6} \][/tex]
