Evaluate : `((3^(4))^(4)xx9^(6))/((27)^(7)xx3^(9))`

Options

A. 3
B. 9
C. `(1)/(3)`
D. `(1)/(9)`

Answer

To evaluate the expression \(\frac{(3^4)^4 \times 9^6}{27^7 \times 3^9}\), we will simplify it step by step.

### Step 1: Rewrite all terms in powers of 3
We know that:
- \(9 = 3^2\)
- \(27 = 3^3\)

So we can rewrite the expression as:
\[ \frac{(3^4)^4 \times (3^2)^6}{(3^3)^7 \times 3^9} \]

### Step 2: Apply the power of a power rule
Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify the powers:
- \((3^4)^4 = 3^{4 \cdot 4} = 3^{16}\)
- \((3^2)^6 = 3^{2 \cdot 6} = 3^{12}\)
- \((3^3)^7 = 3^{3 \cdot 7} = 3^{21}\)

Now the expression becomes:
\[ \frac{3^{16} \times 3^{12}}{3^{21} \times 3^9} \]

### Step 3: Combine the powers in the numerator
Using the rule \(a^m \times a^n = a^{m+n}\), we can combine the powers in the numerator:
\[ 3^{16 + 12} = 3^{28} \]

So now we have:
\[ \frac{3^{28}}{3^{21} \times 3^9} \]

### Step 4: Combine the powers in the denominator
Again using \(a^m \times a^n = a^{m+n}\), we can combine the powers in the denominator:
\[ 3^{21 + 9} = 3^{30} \]

Now the expression simplifies to:
\[ \frac{3^{28}}{3^{30}} \]

### Step 5: Subtract the exponents
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify further:
\[ 3^{28 - 30} = 3^{-2} \]

### Step 6: Convert the negative exponent
A negative exponent means we take the reciprocal:
\[ 3^{-2} = \frac{1}{3^2} \]

### Step 7: Calculate \(3^2\)
Calculating \(3^2\):
\[ 3^2 = 9 \]

Thus, we have:
\[ \frac{1}{3^2} = \frac{1}{9} \]

### Final Answer
The final answer is:
\[ \frac{1}{9} \]

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