`(3a-2b)^3+(2b-5c)^3+(5c-3a)^3`

Answer

To solve the expression \((3a-2b)^3+(2b-5c)^3+(5c-3a)^3\), we can use the identity for the sum of cubes. Here are the steps to arrive at the solution:

### Step 1: Identify the variables
Let:
- \( A = 3a - 2b \)
- \( B = 2b - 5c \)
- \( C = 5c - 3a \)

### Step 2: Check the sum \( A + B + C \)
Now, we need to check if \( A + B + C = 0 \):
\[ A + B + C = (3a - 2b) + (2b - 5c) + (5c - 3a) \]
Combine the terms:
\[ = 3a - 2b + 2b - 5c + 5c - 3a = 0 \]
Thus, \( A + B + C = 0 \).

### Step 3: Use the sum of cubes identity
We can use the identity:
\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - AC - BC) \]
Since \( A + B + C = 0 \), we have:
\[ A^3 + B^3 + C^3 = 3ABC \]

### Step 4: Calculate \( ABC \)
Now, we need to calculate \( ABC \):
\[ ABC = (3a - 2b)(2b - 5c)(5c - 3a) \]

### Step 5: Expand \( ABC \)
To find \( ABC \), we can expand it step by step. First, calculate \( (3a - 2b)(2b - 5c) \):
\[ = 3a(2b) + 3a(-5c) - 2b(2b) - 2b(-5c) = 6ab - 15ac - 4b^2 + 10bc \]
Now multiply this result by \( (5c - 3a) \):
\[ = (6ab - 15ac - 4b^2 + 10bc)(5c - 3a) \]
Distributing each term:
\[ = 6ab(5c) - 6ab(3a) - 15ac(5c) + 15ac(3a) - 4b^2(5c) + 4b^2(3a) + 10bc(5c) - 10bc(3a) \]
Combine like terms to find the final expression for \( ABC \).

### Step 6: Final Expression
After calculating \( ABC \), substitute it back into the equation:
\[ A^3 + B^3 + C^3 = 3ABC \]
This gives us the final result for the original expression.