`(4^(5)xx3^5)/((12)^(5) xx9^2)=`?

Options

A. `3^(2)xx2^5`
B. `(1)/(3^4)`
C. `12`
D. `(1)/(4^3)`

Answer

To solve the expression \((4^{5} \times 3^{5}) / (12^{5} \times 9^{2})\), we will simplify it step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{4^{5} \times 3^{5}}{12^{5} \times 9^{2}} \]

### Step 2: Factor the bases
Next, we can factor \(12\) and \(9\) into their prime factors:
- \(12 = 4 \times 3\)
- \(9 = 3^{2}\)

Thus, we can rewrite \(12^{5}\) and \(9^{2}\):
\[ 12^{5} = (4 \times 3)^{5} = 4^{5} \times 3^{5} \]
\[ 9^{2} = (3^{2})^{2} = 3^{4} \]

### Step 3: Substitute back into the expression
Now, substituting these back into the expression gives us:
\[ \frac{4^{5} \times 3^{5}}{(4^{5} \times 3^{5}) \times 3^{4}} \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ = \frac{4^{5} \times 3^{5}}{4^{5} \times 3^{5} \times 3^{4}} \]
The \(4^{5} \times 3^{5}\) in the numerator and denominator cancels out:
\[ = \frac{1}{3^{4}} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{3^{4}} \]