The value of `(9^(x) (9^(x-1) )^(x) )/( 9^(x+1) . 3^(2x-2) ) { (729^((x)/(3)) )/( 81) }^(-x) div (3^(a) - 2^(3) . 3^(a-2) )/( 3^(a) - 3^(a-1) )` is

Options

A. 9
B. 6
C. 12
D. 16

Answer

To solve the expression

\[ \frac{9^x (9^{x-1})^x}{9^{x+1} \cdot 3^{2x-2}} \cdot \left(\frac{729^{\frac{x}{3}}}{81}\right)^{-x} \div \frac{3^a - 2^3 \cdot 3^{a-2}}{3^a - 3^{a-1}} \]

we will break it down step by step.

### Step 1: Simplify the first part of the expression

The first part is

\[ \frac{9^x (9^{x-1})^x}{9^{x+1} \cdot 3^{2x-2}} \]

We can simplify \( (9^{x-1})^x \) as follows:

\[ (9^{x-1})^x = 9^{x(x-1)} = 9^{x^2 - x} \]

Now, substituting this back into the expression gives:

\[ \frac{9^x \cdot 9^{x^2 - x}}{9^{x+1} \cdot 3^{2x-2}} \]

Using the property of exponents \( a^m \cdot a^n = a^{m+n} \):

\[ = \frac{9^{x + x^2 - x}}{9^{x+1} \cdot 3^{2x-2}} = \frac{9^{x^2}}{9^{x+1} \cdot 3^{2x-2}} \]

### Step 2: Simplify the denominator

Now, we can simplify the denominator:

\[ 9^{x+1} = 9^x \cdot 9^1 = 9^x \cdot 9 \]

Thus, we have:

\[ = \frac{9^{x^2}}{9^x \cdot 9 \cdot 3^{2x-2}} = \frac{9^{x^2}}{9^{x+1} \cdot 3^{2x-2}} \]

### Step 3: Combine the powers of 9

Now, we can combine the powers of 9:

\[ = \frac{9^{x^2}}{9^{x+1}} \cdot \frac{1}{3^{2x-2}} = 9^{x^2 - (x+1)} \cdot 3^{-(2x-2)} = 9^{x^2 - x - 1} \cdot 3^{-(2x-2)} \]

### Step 4: Simplify \( 729^{\frac{x}{3}} \)

Next, we simplify \( \left(\frac{729^{\frac{x}{3}}}{81}\right)^{-x} \):

Since \( 729 = 9^3 \) and \( 81 = 9^2 \):

\[ \frac{729^{\frac{x}{3}}}{81} = \frac{(9^3)^{\frac{x}{3}}}{9^2} = \frac{9^{x}}{9^2} = 9^{x-2} \]

Then, raising this to the power of \(-x\):

\[ (9^{x-2})^{-x} = 9^{-x(x-2)} = 9^{-x^2 + 2x} \]

### Step 5: Combine everything

Now, we can combine everything:

\[ 9^{x^2 - x - 1} \cdot 9^{-x^2 + 2x} = 9^{(x^2 - x - 1) + (-x^2 + 2x)} = 9^{x - 1} \]

### Step 6: Simplify the division part

Now, we need to simplify the division part:

\[ \frac{3^a - 2^3 \cdot 3^{a-2}}{3^a - 3^{a-1}} = \frac{3^a - 8 \cdot 3^{a-2}}{3^a - 3^{a-1}} \]

Factoring out \( 3^{a-2} \) from the numerator:

\[ = \frac{3^{a-2}(3^2 - 8)}{3^{a-1}(3-1)} = \frac{3^{a-2}(9 - 8)}{3^{a-1}(2)} = \frac{3^{a-2}}{3^{a-1}} \cdot \frac{1}{2} = \frac{1}{2 \cdot 3^{1}} = \frac{1}{6} \]

### Step 7: Final expression

Now, we have:

\[ 9^{x-1} \cdot 6 \]

Thus, the final value of the entire expression is:

\[ 6 \cdot 9^{x-1} \]