If `3x+2y=14` and `xy=8`, then `(27x^3+8y^3)` is equal to:

Options

A. 816
B. 728
C. 832
D. 1228

Answer

To solve the problem, we need to find the value of \( 27x^3 + 8y^3 \) given the equations \( 3x + 2y = 14 \) and \( xy = 8 \).

### Step 1: Use the identity for the sum of cubes
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
In our case, let \( a = 3x \) and \( b = 2y \). Then,
\[ 27x^3 + 8y^3 = (3x)^3 + (2y)^3 = (3x + 2y)((3x)^2 - (3x)(2y) + (2y)^2) \]

### Step 2: Substitute the known values
From the problem, we know:
\[ 3x + 2y = 14 \]
Now we need to calculate \( (3x)^2 - (3x)(2y) + (2y)^2 \).

### Step 3: Calculate each term
1. Calculate \( (3x)^2 \):
\[ (3x)^2 = 9x^2 \]
2. Calculate \( (2y)^2 \):
\[ (2y)^2 = 4y^2 \]
3. Calculate \( (3x)(2y) \):
\[ (3x)(2y) = 6xy \]
Since \( xy = 8 \), we have:
\[ 6xy = 6 \times 8 = 48 \]

### Step 4: Combine the terms
Now we can substitute back into our expression:
\[ (3x)^2 - (3x)(2y) + (2y)^2 = 9x^2 - 48 + 4y^2 \]

### Step 5: Express \( 9x^2 + 4y^2 \) in terms of \( xy \)
We know that:
\[ (3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2 \]
Expanding this gives:
\[ 14^2 = 9x^2 + 4y^2 + 12xy \]
Calculating \( 14^2 \):
\[ 196 = 9x^2 + 4y^2 + 12 \times 8 \]
\[ 196 = 9x^2 + 4y^2 + 96 \]
Thus,
\[ 9x^2 + 4y^2 = 196 - 96 = 100 \]

### Step 6: Substitute back into the expression
Now we substitute back:
\[ (3x)^2 - (3x)(2y) + (2y)^2 = 100 - 48 = 52 \]

### Step 7: Calculate \( 27x^3 + 8y^3 \)
Now we can find \( 27x^3 + 8y^3 \):
\[ 27x^3 + 8y^3 = (3x + 2y)(9x^2 - 6xy + 4y^2) = 14 \times 52 \]
Calculating this gives:
\[ 27x^3 + 8y^3 = 728 \]

### Final Answer
Thus, the value of \( 27x^3 + 8y^3 \) is \( \boxed{728} \).