If `x+ 5y= 10`, find the value of `x^(3) + 125y^(3) + 150xy - 1000`

Answer

To solve the problem, we need to find the value of \( x^3 + 125y^3 + 150xy - 1000 \) given that \( x + 5y = 10 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + 5y = 10 \]

2. **Cube both sides of the equation**:
\[ (x + 5y)^3 = 10^3 \]
This simplifies to:
\[ (x + 5y)^3 = 1000 \]

3. **Use the binomial expansion formula**:
The formula for \( (a + b)^3 \) is:
\[ a^3 + b^3 + 3ab(a + b) \]
Here, let \( a = x \) and \( b = 5y \). Therefore, we have:
\[ x^3 + (5y)^3 + 3(x)(5y)(x + 5y) = 1000 \]

4. **Calculate \( (5y)^3 \)**:
\[ (5y)^3 = 125y^3 \]
So the equation becomes:
\[ x^3 + 125y^3 + 15xy(x + 5y) = 1000 \]

5. **Substitute \( x + 5y \) with 10**:
Since \( x + 5y = 10 \), we can substitute this into the equation:
\[ x^3 + 125y^3 + 15xy(10) = 1000 \]
This simplifies to:
\[ x^3 + 125y^3 + 150xy = 1000 \]

6. **Rearranging the equation**:
Now, we need to find \( x^3 + 125y^3 + 150xy - 1000 \):
\[ x^3 + 125y^3 + 150xy - 1000 = 0 \]

### Final Answer:
Thus, the value of \( x^3 + 125y^3 + 150xy - 1000 \) is:
\[ \boxed{0} \]