Factorise :
`(i) (x-y)^(3)+(y-z)^(3)+(z-x)^(3)`
`(ii) (x-2y)^(3)+(2y-4z)^(3)+(4z-x)^(3)`
`(iv) (3sqrt(2)a-5sqrt(3)b)^(3)+(5sqrt(3)b-7sqrt(5)c)^(3)+(7sqrt(5)c-3sqrt(2)a)^(3)`.

Answer

To factorize the given expressions, we will use the identity that states if \( a + b + c = 0 \), then:

\[ a^3 + b^3 + c^3 = 3abc \]

Let's solve each part step by step.

### Part (i): Factorize \( (x-y)^3 + (y-z)^3 + (z-x)^3 \)

1. **Identify \( a, b, c \)**:
- Let \( a = x - y \)
- Let \( b = y - z \)
- Let \( c = z - x \)

2. **Check if \( a + b + c = 0 \)**:
\[ a + b + c = (x - y) + (y - z) + (z - x) = 0 \]
Since the sum is zero, we can use the identity.

3. **Apply the identity**:
\[ (x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x) \]

### Final Answer for Part (i):
\[ (x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x) \]

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### Part (ii): Factorize \( (x-2y)^3 + (2y-4z)^3 + (4z-x)^3 \)

1. **Identify \( a, b, c \)**:
- Let \( a = x - 2y \)
- Let \( b = 2y - 4z \)
- Let \( c = 4z - x \)

2. **Check if \( a + b + c = 0 \)**:
\[ a + b + c = (x - 2y) + (2y - 4z) + (4z - x) = 0 \]
The sum is zero, so we can use the identity.

3. **Apply the identity**:
\[ (x - 2y)^3 + (2y - 4z)^3 + (4z - x)^3 = 3(x - 2y)(2y - 4z)(4z - x) \]

### Final Answer for Part (ii):
\[ (x - 2y)^3 + (2y - 4z)^3 + (4z - x)^3 = 3(x - 2y)(2y - 4z)(4z - x) \]

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### Part (iii): Factorize \( (3\sqrt{2}a - 5\sqrt{3}b)^3 + (5\sqrt{3}b - 7\sqrt{5}c)^3 + (7\sqrt{5}c - 3\sqrt{2}a)^3 \)

1. **Identify \( a, b, c \)**:
- Let \( a = 3\sqrt{2}a - 5\sqrt{3}b \)
- Let \( b = 5\sqrt{3}b - 7\sqrt{5}c \)
- Let \( c = 7\sqrt{5}c - 3\sqrt{2}a \)

2. **Check if \( a + b + c = 0 \)**:
\[ a + b + c = (3\sqrt{2}a - 5\sqrt{3}b) + (5\sqrt{3}b - 7\sqrt{5}c) + (7\sqrt{5}c - 3\sqrt{2}a) = 0 \]
The sum is zero, thus we can apply the identity.

3. **Apply the identity**:
\[ (3\sqrt{2}a - 5\sqrt{3}b)^3 + (5\sqrt{3}b - 7\sqrt{5}c)^3 + (7\sqrt{5}c - 3\sqrt{2}a)^3 = 3(3\sqrt{2}a - 5\sqrt{3}b)(5\sqrt{3}b - 7\sqrt{5}c)(7\sqrt{5}c - 3\sqrt{2}a) \]

### Final Answer for Part (iii):
\[ (3\sqrt{2}a - 5\sqrt{3}b)^3 + (5\sqrt{3}b - 7\sqrt{5}c)^3 + (7\sqrt{5}c - 3\sqrt{2}a)^3 = 3(3\sqrt{2}a - 5\sqrt{3}b)(5\sqrt{3}b - 7\sqrt{5}c)(7\sqrt{5}c - 3\sqrt{2}a) \]

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