`(5a+2b)^(3)` simplify

Answer

To simplify \((5a + 2b)^3\), we will use the binomial expansion formula, specifically the identity for the cube of a binomial:

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

### Step-by-step Solution:

1. **Identify \(a\) and \(b\)**:
- In our case, let \(a = 5a\) and \(b = 2b\).

2. **Apply the binomial expansion formula**:
- Substitute \(a\) and \(b\) into the formula:
\[ (5a + 2b)^3 = (5a)^3 + (2b)^3 + 3(5a)(2b)(5a + 2b) \]

3. **Calculate \((5a)^3\)**:
- \((5a)^3 = 125a^3\).

4. **Calculate \((2b)^3\)**:
- \((2b)^3 = 8b^3\).

5. **Calculate \(3(5a)(2b)\)**:
- \(3(5a)(2b) = 30ab\).

6. **Combine all parts**:
- Now substitute back into the expanded form:
\[ (5a + 2b)^3 = 125a^3 + 8b^3 + 30ab(5a + 2b) \]

7. **Distribute \(30ab\) into \((5a + 2b)\)**:
- \(30ab(5a) + 30ab(2b) = 150a^2b + 60ab^2\).

8. **Final expression**:
- Combine all the terms:
\[ (5a + 2b)^3 = 125a^3 + 8b^3 + 150a^2b + 60ab^2 \]

### Final Answer:
\[ (5a + 2b)^3 = 125a^3 + 8b^3 + 150a^2b + 60ab^2 \]