Factorise:`(a+b)^3-(a-b)^3`

Explanation

To factorise the expression \((a+b)^3 - (a-b)^3\), we can use the difference of cubes formula.

### Step-by-Step Solution:

1. **Identify the Cubes**:
We have two cubes in the expression:
- Let \(x = (a+b)\) and \(y = (a-b)\).
- Thus, we can rewrite the expression as \(x^3 - y^3\).

2. **Apply the Difference of Cubes Formula**:
The difference of cubes can be factored using the formula:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
Applying this to our expression:
\[ (a+b)^3 - (a-b)^3 = ((a+b) - (a-b))((a+b)^2 + (a+b)(a-b) + (a-b)^2) \]

3. **Simplify \(x - y\)**:
Calculate \(x - y\):
\[ (a+b) - (a-b) = a + b - a + b = 2b \]

4. **Calculate \(x^2 + xy + y^2\)**:
Now we need to calculate each term:
- \(x^2 = (a+b)^2 = a^2 + 2ab + b^2\)
- \(y^2 = (a-b)^2 = a^2 - 2ab + b^2\)
- \(xy = (a+b)(a-b) = a^2 - b^2\)

Now substitute these into the expression:
\[ (a+b)^2 + (a+b)(a-b) + (a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - b^2) + (a^2 - 2ab + b^2) \]

5. **Combine Like Terms**:
Combine all the terms:
\[ = a^2 + 2ab + b^2 + a^2 - b^2 + a^2 - 2ab + b^2 \]
Simplifying this gives:
\[ = 3a^2 + b^2 \]

6. **Final Factorization**:
Now we can combine everything:
\[ (a+b)^3 - (a-b)^3 = 2b(3a^2 + b^2) \]

### Final Answer:
Thus, the factorized form of \((a+b)^3 - (a-b)^3\) is:
\[ 2b(3a^2 + b^2) \]