Find the value of `(3^(n)xx3^(2n+1))/(9^(n)xx3^(n-1))`.

Options

A. `8`
B. `1`
C. `27`
D. `9`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{3^n \times 3^{2n+1}}{9^n \times 3^{n-1}} \]

### Step 2: Substitute \(9\) with \(3^2\)
Since \(9\) can be expressed as \(3^2\), we can rewrite \(9^n\) as \((3^2)^n\):
\[ 9^n = (3^2)^n = 3^{2n} \]
Now, substitute this back into the expression:
\[ \frac{3^n \times 3^{2n+1}}{3^{2n} \times 3^{n-1}} \]

### Step 3: Simplify the numerator
Using the property of exponents \(a^m \times a^n = a^{m+n}\), we can simplify the numerator:
\[ 3^n \times 3^{2n+1} = 3^{n + (2n + 1)} = 3^{3n + 1} \]

### Step 4: Simplify the denominator
Now simplify the denominator:
\[ 3^{2n} \times 3^{n-1} = 3^{2n + (n - 1)} = 3^{3n - 1} \]

### Step 5: Combine the simplified numerator and denominator
Now, we can rewrite the expression as:
\[ \frac{3^{3n + 1}}{3^{3n - 1}} \]

### Step 6: Apply the quotient rule of exponents
Using the property \(a^m / a^n = a^{m-n}\):
\[ 3^{(3n + 1) - (3n - 1)} = 3^{3n + 1 - 3n + 1} = 3^{2} \]

### Step 7: Calculate the final value
Now, we can calculate \(3^2\):
\[ 3^2 = 9 \]

Thus, the value of the expression \((3^n \times 3^{2n+1}) / (9^n \times 3^{n-1})\) is \(\boxed{9}\).

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