`3^(5)x^3y^1 div 9x^(-1)y^(2)=`______

Options

A. 27xy
B. 27 `x^4y^(-1)`
C. 27xz
D. 27xyz

Answer

To solve the expression \( \frac{3^5 x^3 y^1}{9 x^{-1} y^2} \), we will follow these steps:

### Step 1: Rewrite the denominator
First, we need to rewrite the denominator \( 9 \) in terms of powers of \( 3 \):
\[ 9 = 3^2 \]
So, we can rewrite the expression as:
\[ \frac{3^5 x^3 y^1}{3^2 x^{-1} y^2} \]

### Step 2: Apply the properties of exponents
Now, we can simplify the expression using the properties of exponents. We can separate the terms:
\[ \frac{3^5}{3^2} \cdot \frac{x^3}{x^{-1}} \cdot \frac{y^1}{y^2} \]

### Step 3: Simplify each part
1. For the \( 3 \) terms:
\[ \frac{3^5}{3^2} = 3^{5-2} = 3^3 \]

2. For the \( x \) terms:
\[ \frac{x^3}{x^{-1}} = x^{3 - (-1)} = x^{3 + 1} = x^4 \]

3. For the \( y \) terms:
\[ \frac{y^1}{y^2} = y^{1-2} = y^{-1} \]

### Step 4: Combine the results
Now, we can combine all the simplified parts:
\[ 3^3 \cdot x^4 \cdot y^{-1} \]

### Step 5: Write the final answer
Calculating \( 3^3 \):
\[ 3^3 = 27 \]
Thus, the final expression is:
\[ 27 x^4 y^{-1} \]

### Final Answer:
\[ \frac{3^5 x^3 y^1}{9 x^{-1} y^2} = 27 x^4 y^{-1} \]