If `2x- 3y= 10 and xy= 16`, find the value of `8x^(3)- 27y^(3)`

Answer

To solve the problem, we need to find the value of \( 8x^3 - 27y^3 \) given the equations \( 2x - 3y = 10 \) and \( xy = 16 \).

### Step-by-Step Solution:

1. **Use the identity for cubes**:
We start by recognizing that \( 8x^3 - 27y^3 \) can be expressed using the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + b^2 + ab) \]
Here, we can let \( a = 2x \) and \( b = 3y \). Thus, we rewrite:
\[ 8x^3 - 27y^3 = (2x)^3 - (3y)^3 = (2x - 3y)((2x)^2 + (3y)^2 + (2x)(3y)) \]

2. **Substituting the known values**:
From the problem, we know that \( 2x - 3y = 10 \). So we can substitute this into our expression:
\[ 8x^3 - 27y^3 = 10 \left((2x)^2 + (3y)^2 + (2x)(3y)\right) \]

3. **Calculating \( (2x)^2 + (3y)^2 + (2x)(3y) \)**:
We need to calculate \( (2x)^2 \), \( (3y)^2 \), and \( (2x)(3y) \):
- \( (2x)^2 = 4x^2 \)
- \( (3y)^2 = 9y^2 \)
- \( (2x)(3y) = 6xy \)

Thus, we have:
\[ (2x)^2 + (3y)^2 + (2x)(3y) = 4x^2 + 9y^2 + 6xy \]

4. **Finding \( 4x^2 + 9y^2 \)**:
We can find \( 4x^2 + 9y^2 \) using the equation \( (2x - 3y)^2 = 100 \):
\[ (2x - 3y)^2 = 4x^2 + 9y^2 - 12xy \]
Substituting \( 2x - 3y = 10 \) gives:
\[ 100 = 4x^2 + 9y^2 - 12xy \]
We know \( xy = 16 \), so:
\[ 100 = 4x^2 + 9y^2 - 12 \cdot 16 \]
This simplifies to:
\[ 100 = 4x^2 + 9y^2 - 192 \]
Rearranging gives:
\[ 4x^2 + 9y^2 = 100 + 192 = 292 \]

5. **Putting it all together**:
Now substitute back into the equation:
\[ 8x^3 - 27y^3 = 10 \left(292 + 96\right) \]
Where \( 6xy = 6 \cdot 16 = 96 \).

6. **Final calculation**:
Thus:
\[ 8x^3 - 27y^3 = 10 \cdot 388 = 3880 \]

### Final Answer:
\[ \boxed{3880} \]