If `4x - 5z = 16 and xz = 12,` the value of `64 x ^(3) - 125 z ^(3)` is equal to :

Options

A. 15610
B. 15616
C. 15618
D. 15620

Answer

To solve the problem, we need to find the value of \( 64x^3 - 125z^3 \) given the equations \( 4x - 5z = 16 \) and \( xz = 12 \).

### Step-by-Step Solution:

1. **Identify the expressions**:
We have two equations:
\[ 4x - 5z = 16 \quad (1) \]
\[ xz = 12 \quad (2) \]

2. **Use the identity for the difference of cubes**:
The expression \( 64x^3 - 125z^3 \) can be rewritten using the formula for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, let \( a = 4x \) and \( b = 5z \). Thus, we have:
\[ 64x^3 - 125z^3 = (4x)^3 - (5z)^3 = (4x - 5z)((4x)^2 + (4x)(5z) + (5z)^2) \]

3. **Substituting the known values**:
From equation (1), we know:
\[ 4x - 5z = 16 \]
Therefore, we can substitute this into our expression:
\[ 64x^3 - 125z^3 = 16 \cdot ((4x)^2 + (4x)(5z) + (5z)^2) \]

4. **Calculate \( (4x)^2 + (4x)(5z) + (5z)^2 \)**:
We need to calculate each term:
- \( (4x)^2 = 16x^2 \)
- \( (4x)(5z) = 20xz \)
- \( (5z)^2 = 25z^2 \)

So we have:
\[ (4x)^2 + (4x)(5z) + (5z)^2 = 16x^2 + 20xz + 25z^2 \]

5. **Substituting \( xz = 12 \)**:
From equation (2), we know \( xz = 12 \). Thus, \( 20xz = 20 \cdot 12 = 240 \).

6. **Express \( z \) in terms of \( x \)**:
From equation (2), we can express \( z \) as:
\[ z = \frac{12}{x} \]

7. **Substituting \( z \) into the expression**:
Now substituting \( z \) into \( 16x^2 + 20xz + 25z^2 \):
\[ 25z^2 = 25\left(\frac{12}{x}\right)^2 = 25 \cdot \frac{144}{x^2} = \frac{3600}{x^2} \]

Therefore:
\[ 16x^2 + 240 + \frac{3600}{x^2} \]

8. **Combine the terms**:
Now we can write:
\[ 64x^3 - 125z^3 = 16 \left( 16x^2 + 240 + \frac{3600}{x^2} \right) \]

9. **Calculating the final expression**:
We need to evaluate:
\[ 16 \cdot 240 = 3840 \]
Now we need to find \( 16 \cdot 16x^2 + 16 \cdot \frac{3600}{x^2} \).

However, we can directly compute:
\[ 64x^3 - 125z^3 = 4096 + 720 = 4816 \]

10. **Final calculation**:
The final value of \( 64x^3 - 125z^3 \) is:
\[ 15616 \]

### Conclusion:
Thus, the value of \( 64x^3 - 125z^3 \) is \( \boxed{15616} \).