Find the value of `((243)^((n)/(5))xx3^(2n+1))/(9^(n)xx3^(n-1))`

Options

A. `3`
B. `9`
C. `27`
D. `4`

Answer

To solve the expression \(\frac{(243)^{\frac{n}{5}} \times 3^{2n+1}}{9^{n} \times 3^{n-1}}\), we will simplify it step by step.

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

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

### Step 2: Simplify the powers
Now, apply the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[ (3^5)^{\frac{n}{5}} = 3^{5 \cdot \frac{n}{5}} = 3^n \]
\[ (3^2)^{n} = 3^{2n} \]

Substituting these back into the expression gives:
\[ \frac{3^n \times 3^{2n+1}}{3^{2n} \times 3^{n-1}} \]

### Step 3: Combine the powers in the numerator
Using the property \(a^m \times a^n = a^{m+n}\):
\[ 3^n \times 3^{2n+1} = 3^{n + (2n + 1)} = 3^{3n + 1} \]

### Step 4: Combine the powers in the denominator
Similarly, for the denominator:
\[ 3^{2n} \times 3^{n-1} = 3^{(2n + (n - 1))} = 3^{3n - 1} \]

### Step 5: Simplify the entire expression
Now, we can simplify the fraction:
\[ \frac{3^{3n + 1}}{3^{3n - 1}} = 3^{(3n + 1) - (3n - 1)} = 3^{3n + 1 - 3n + 1} = 3^{2} \]

### Step 6: Final answer
Thus, the value of the expression is:
\[ 3^2 = 9 \]