Simplify : `((x)/(3)+(y)/(5))^(3)-((x)/(3)-(y)/(5))^(3)`

Options

A. `(2x)/(5)((x^(2))/(3)-(y^(2))/(25))`
B. `(2y)/(5)((x^(2))/(3)-(y^(2))/(25))`
C. `(2x)/(5)((x^(2))/(3)+(y^(2))/(25))`
D. `(2y)/(5)((x^(2))/(3)+(y^(2))/(25))`

Answer

To simplify the expression \(\left(\frac{x}{3} + \frac{y}{5}\right)^{3} - \left(\frac{x}{3} - \frac{y}{5}\right)^{3}\), we can use the identity for the difference of cubes.

### Step-by-step solution:

1. **Identify the terms**: Let \( a = \frac{x}{3} \) and \( b = \frac{y}{5} \). Then the expression can be rewritten as:
\[ (a + b)^{3} - (a - b)^{3} \]

2. **Use the difference of cubes formula**: The formula for the difference of cubes states that:
\[ A^{3} - B^{3} = (A - B)(A^{2} + AB + B^{2}) \]
Here, \( A = a + b \) and \( B = a - b \).

3. **Calculate \( A - B \)**:
\[ A - B = (a + b) - (a - b) = 2b \]

4. **Calculate \( A^{2} + AB + B^{2} \)**:
- First, calculate \( A^{2} \):
\[ A^{2} = (a + b)^{2} = a^{2} + 2ab + b^{2} \]
- Next, calculate \( AB \):
\[ AB = (a + b)(a - b) = a^{2} - b^{2} \]
- Finally, calculate \( B^{2} \):
\[ B^{2} = (a - b)^{2} = a^{2} - 2ab + b^{2} \]
- Now combine these:
\[ A^{2} + AB + B^{2} = (a^{2} + 2ab + b^{2}) + (a^{2} - b^{2}) + (a^{2} - 2ab + b^{2}) \]
\[ = 3a^{2} + 3b^{2} \]

5. **Combine the results**:
\[ (a + b)^{3} - (a - b)^{3} = (2b)(3a^{2} + 3b^{2}) = 6b(a^{2} + b^{2}) \]

6. **Substitute back \( a \) and \( b \)**:
\[ = 6 \left(\frac{y}{5}\right) \left(\left(\frac{x}{3}\right)^{2} + \left(\frac{y}{5}\right)^{2}\right) \]
\[ = 6 \cdot \frac{y}{5} \cdot \left(\frac{x^{2}}{9} + \frac{y^{2}}{25}\right) \]

7. **Simplify the expression**:
\[ = \frac{6y}{5} \cdot \left(\frac{x^{2}}{9} + \frac{y^{2}}{25}\right) \]
\[ = \frac{6y}{5} \cdot \left(\frac{25x^{2} + 9y^{2}}{225}\right) \]
\[ = \frac{6y(25x^{2} + 9y^{2})}{1125} \]

### Final Result:
\[ \frac{6y(25x^{2} + 9y^{2})}{1125} \]