If `(9^(n)xx3^(5)xx(27)^3)/(3xx(81)^4)=27`, then the value of n is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To solve the equation

\[ \frac{9^n \times 3^5 \times (27)^3}{3 \times (81)^4} = 27, \]

we will simplify and find the value of \( n \).

### Step 1: Rewrite all numbers as powers of 3
First, we rewrite all the numbers in the equation using base 3:

- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
- \( 81 = 3^4 \)

So we can rewrite the expression:

\[ 9^n = (3^2)^n = 3^{2n}, \]
\[ 27^3 = (3^3)^3 = 3^9, \]
\[ 81^4 = (3^4)^4 = 3^{16}. \]

Now substituting these into the equation gives:

\[ \frac{3^{2n} \times 3^5 \times 3^9}{3 \times 3^{16}} = 27. \]

### Step 2: Simplify the left side
Combine the powers of 3 in the numerator:

\[ 3^{2n + 5 + 9} = 3^{2n + 14}. \]

The denominator simplifies to:

\[ 3^{1 + 16} = 3^{17}. \]

So the left side becomes:

\[ \frac{3^{2n + 14}}{3^{17}} = 3^{(2n + 14) - 17} = 3^{2n - 3}. \]

### Step 3: Set the equation
Now we have:

\[ 3^{2n - 3} = 27. \]

Since \( 27 = 3^3 \), we can rewrite the equation as:

\[ 3^{2n - 3} = 3^3. \]

### Step 4: Equate the exponents
Since the bases are the same, we can equate the exponents:

\[ 2n - 3 = 3. \]

### Step 5: Solve for \( n \)
Now, solve for \( n \):

\[ 2n = 3 + 3, \]
\[ 2n = 6, \]
\[ n = \frac{6}{2} = 3. \]

### Final Answer
The value of \( n \) is \( 3 \).

---