Simplify: `(((6)/(5))^(2)xx x^(8)xx y^(7))div(2^(10)xx (x/3)^(4)xx (y/3)^(3))`

Options

A. `(3^9)/(5^(2)xx 2^(8))(xy)^(4)`
B. `(5^(2)xx 2^(5))/(3^9)(xy)^(4)`
C. `((3)/(10))^(4)(xy)^4`
D. `(3^9)/(5^(2)xx 2^(8))x^(12)y^(8)`

Answer

To simplify the expression \(\frac{\left(\frac{6}{5}\right)^2 \cdot x^8 \cdot y^7}{2^{10} \cdot \left(\frac{x}{3}\right)^4 \cdot \left(\frac{y}{3}\right)^3}\), we will follow these steps:

### Step 1: Rewrite the expression
First, we rewrite the expression clearly:
\[ \frac{\left(\frac{6}{5}\right)^2 \cdot x^8 \cdot y^7}{2^{10} \cdot \left(\frac{x^4}{3^4}\right) \cdot \left(\frac{y^3}{3^3}\right)} \]

### Step 2: Simplify the denominator
The denominator can be simplified as follows:
\[ 2^{10} \cdot \left(\frac{x^4}{3^4}\right) \cdot \left(\frac{y^3}{3^3}\right) = 2^{10} \cdot \frac{x^4 \cdot y^3}{3^7} \]
Thus, the expression becomes:
\[ \frac{\left(\frac{6}{5}\right)^2 \cdot x^8 \cdot y^7 \cdot 3^7}{2^{10} \cdot x^4 \cdot y^3} \]

### Step 3: Multiply by the reciprocal
Now, we can multiply by the reciprocal of the denominator:
\[ = \left(\frac{6}{5}\right)^2 \cdot x^8 \cdot y^7 \cdot \frac{3^7}{2^{10} \cdot x^4 \cdot y^3} \]

### Step 4: Combine the terms
Now we can combine the terms:
\[ = \frac{6^2 \cdot 3^7 \cdot x^{8-4} \cdot y^{7-3}}{5^2 \cdot 2^{10}} \]
This simplifies to:
\[ = \frac{36 \cdot 3^7 \cdot x^4 \cdot y^4}{25 \cdot 2^{10}} \]

### Step 5: Final expression
Thus, the final simplified expression is:
\[ = \frac{36 \cdot 3^7 \cdot x^4 \cdot y^4}{25 \cdot 1024} \]

### Step 6: Further simplification (if necessary)
Since \(2^{10} = 1024\), we can write:
\[ = \frac{36 \cdot 3^7 \cdot x^4 \cdot y^4}{25600} \]

### Final Result
The simplified form of the expression is:
\[ \frac{36 \cdot 3^7 \cdot x^4 \cdot y^4}{25600} \]