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

Options

A. 3000
B. 3200
C. 3880
D. 2556

Answer

To find the value of \(8x^3 - 27y^3\) given the equations \(2x - 3y = 10\) and \(xy = 16\), we can use the identity for the difference of cubes:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

In our case, we can set \(a = 2x\) and \(b = 3y\). Therefore, we can rewrite \(8x^3 - 27y^3\) as:

\[ 8x^3 - 27y^3 = (2x)^3 - (3y)^3 \]

Using the identity, we have:

\[ (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2) \]

### Step 1: Calculate \(2x - 3y\)

From the problem, we know:

\[ 2x - 3y = 10 \]

### Step 2: Calculate \((2x)^2 + (2x)(3y) + (3y)^2\)

Now we need to calculate \((2x)^2 + (2x)(3y) + (3y)^2\):

1. \((2x)^2 = 4x^2\)
2. \((2x)(3y) = 6xy\)
3. \((3y)^2 = 9y^2\)

So, we have:

\[ (2x)^2 + (2x)(3y) + (3y)^2 = 4x^2 + 6xy + 9y^2 \]

### Step 3: Substitute \(xy\)

We know \(xy = 16\), so:

\[ 6xy = 6 \times 16 = 96 \]

### Step 4: Express \(4x^2 + 9y^2\) in terms of \(x\) and \(y\)

To find \(4x^2 + 9y^2\), we can use the identity:

\[ (2x - 3y)^2 = 4x^2 - 12xy + 9y^2 \]

Substituting \(2x - 3y = 10\):

\[ 10^2 = 4x^2 - 12(16) + 9y^2 \]

This simplifies to:

\[ 100 = 4x^2 - 192 + 9y^2 \]

Rearranging gives:

\[ 4x^2 + 9y^2 = 100 + 192 = 292 \]

### Step 5: Combine results

Now we can substitute back into our expression:

\[ (2x - 3y)(4x^2 + 6xy + 9y^2) = 10(292 + 96) = 10 \times 388 = 3880 \]

### Final Answer

Thus, the value of \(8x^3 - 27y^3\) is:

\[ \boxed{3880} \]