Simplify and write in exponential form : `(12^(4)xx9^(4)xx4)/(6^(3)xx8^(2)xx27)`

Answer

To simplify the expression \((12^{4} \times 9^{4} \times 4) / (6^{3} \times 8^{2} \times 27)\) and write it in exponential form, follow these steps:

### Step 1: Rewrite each term in terms of prime factors
- \(12 = 2^{2} \times 3^{1}\)
- \(9 = 3^{2}\)
- \(4 = 2^{2}\)
- \(6 = 2^{1} \times 3^{1}\)
- \(8 = 2^{3}\)
- \(27 = 3^{3}\)

### Step 2: Substitute the prime factorization into the expression
\[ = \frac{(2^{2} \times 3^{1})^{4} \times (3^{2})^{4} \times (2^{2})}{(2^{1} \times 3^{1})^{3} \times (2^{3})^{2} \times (3^{3})} \]

### Step 3: Apply the power of a product rule
\[ = \frac{2^{8} \times 3^{4} \times 2^{2}}{2^{3} \times 3^{3} \times 2^{6} \times 3^{3}} \]

### Step 4: Combine the powers of the same base in the numerator
\[ = \frac{2^{8 + 2} \times 3^{4}}{2^{3 + 6} \times 3^{3 + 3}} \]
\[ = \frac{2^{10} \times 3^{4}}{2^{9} \times 3^{6}} \]

### Step 5: Apply the quotient rule for exponents
\[ = 2^{10 - 9} \times 3^{4 - 6} \]
\[ = 2^{1} \times 3^{-2} \]

### Step 6: Rewrite negative exponents
\[ = \frac{2^{1}}{3^{2}} \]

### Final Answer
Thus, the simplified expression in exponential form is:
\[ \frac{2}{3^{2}} \]

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