The value of `((243)^(n/5). 3^(2n+1))/(9^(n).3^(n-1))` is:

Options

A. 1
B. 9
C. 3
D. `3^(n)`

Answer

To solve the expression \(\frac{(243)^{(n/5)} \cdot 3^{(2n+1)}}{9^n \cdot 3^{(n-1)}}\), we can follow these steps:

### Step 1: Rewrite the bases
First, we can express \(243\) and \(9\) in terms of base \(3\):
- \(243 = 3^5\)
- \(9 = 3^2\)

So, we can rewrite the expression as:
\[ \frac{(3^5)^{(n/5)} \cdot 3^{(2n+1)}}{(3^2)^n \cdot 3^{(n-1)}} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule, \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (3^5)^{(n/5)} = 3^{(5 \cdot n/5)} = 3^n \]
\[ (3^2)^n = 3^{(2n)} \]

Now, substituting these back into the expression gives us:
\[ \frac{3^n \cdot 3^{(2n+1)}}{3^{(2n)} \cdot 3^{(n-1)}} \]

### Step 3: Combine the powers in the numerator
In the numerator, we can combine the powers of \(3\):
\[ 3^n \cdot 3^{(2n+1)} = 3^{n + 2n + 1} = 3^{(3n + 1)} \]

### Step 4: Combine the powers in the denominator
In the denominator, we can also combine the powers of \(3\):
\[ 3^{(2n)} \cdot 3^{(n-1)} = 3^{(2n + n - 1)} = 3^{(3n - 1)} \]

### Step 5: Simplify the expression
Now we can simplify the entire expression:
\[ \frac{3^{(3n + 1)}}{3^{(3n - 1)}} \]
Using the quotient rule for exponents, \(\frac{a^m}{a^n} = a^{m-n}\):
\[ 3^{(3n + 1) - (3n - 1)} = 3^{(3n + 1 - 3n + 1)} = 3^{2} \]

### Final Result
Thus, the value of the expression is:
\[ 3^2 = 9 \]