Evaluate `(2x-3y+5)^(3)`.

Answer

To evaluate the expression \((2x - 3y + 5)^3\), we can use the formula for the cube of a binomial. However, since we have three terms, we will treat \(2x - 3y\) as one term (let's call it \(a\)) and \(5\) as another term (let's call it \(b\)).

### Step-by-step Solution:

1. **Identify the terms**:
Let \(a = 2x - 3y\) and \(b = 5\).

2. **Use the formula for the cube of a sum**:
The formula for \((a + b)^3\) is:
\[ a^3 + b^3 + 3a^2b + 3ab^2 \]

3. **Calculate \(a^3\)**:
\[ a^3 = (2x - 3y)^3 \]
Using the binomial expansion:
\[ (p - q)^3 = p^3 - 3p^2q + 3pq^2 - q^3 \]
Here, \(p = 2x\) and \(q = 3y\):
\[ = (2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3 \]
\[ = 8x^3 - 3 \cdot 4x^2 \cdot 3y + 3 \cdot 2x \cdot 9y^2 - 27y^3 \]
\[ = 8x^3 - 36x^2y + 54xy^2 - 27y^3 \]

4. **Calculate \(b^3\)**:
\[ b^3 = 5^3 = 125 \]

5. **Calculate \(3a^2b\)**:
\[ 3a^2b = 3(2x - 3y)^2 \cdot 5 \]
First, calculate \((2x - 3y)^2\):
\[ = (2x)^2 - 2(2x)(3y) + (3y)^2 = 4x^2 - 12xy + 9y^2 \]
Now multiply by \(5\):
\[ 3(4x^2 - 12xy + 9y^2) \cdot 5 = 15(4x^2 - 12xy + 9y^2) = 60x^2 - 180xy + 135y^2 \]

6. **Calculate \(3ab^2\)**:
\[ 3ab^2 = 3(2x - 3y)(5^2) = 3(2x - 3y) \cdot 25 = 75(2x - 3y) = 150x - 225y \]

7. **Combine all the terms**:
Now, we combine all the results:
\[ (2x - 3y + 5)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]
\[ = (8x^3 - 36x^2y + 54xy^2 - 27y^3) + 125 + (60x^2 - 180xy + 135y^2) + (150x - 225y) \]

8. **Combine like terms**:
\[ = 8x^3 + (60x^2 - 36x^2y) + (150x - 180xy + 54xy^2 + 135y^2) - 27y^3 + 125 \]
\[ = 8x^3 - 27y^3 + 60x^2 - 36xy + 54xy^2 + 150x - 225y + 125 \]

### Final Answer:
\[ (2x - 3y + 5)^3 = 8x^3 - 27y^3 + 60x^2 - 36xy + 54xy^2 + 150x - 225y + 125 \]