Without actually calculating the cube find the value of the following :
`(i) (-9)^(3)+(4)^(3)+(5)^(3) " " (ii) (-18)^(3)+(9)^(3)+(9)^(3) " " (iii) (16)^(3)+(1)^(3)+(-17)^(3)`
`(iv) (8)^(3)+(3)^(3)+(-11)^(3)`.

Answer

To solve the given problems without calculating the cubes directly, we will use the identity for the sum of cubes. The identity states:

\[ a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + ac + bc) + 3abc \]

From this identity, if \(a + b + c = 0\), then:

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

Now, let's apply this to each part of the question.

### (i) \((-9)^3 + (4)^3 + (5)^3\)

1. **Identify \(a\), \(b\), and \(c\)**:
- \(a = -9\), \(b = 4\), \(c = 5\)

2. **Calculate \(a + b + c\)**:
\[ a + b + c = -9 + 4 + 5 = 0 \]

3. **Since \(a + b + c = 0\)**, we use the identity:
\[ (-9)^3 + (4)^3 + (5)^3 = 3(-9)(4)(5) \]

4. **Calculate \(3abc\)**:
\[ 3(-9)(4)(5) = 3 \times -9 \times 20 = -540 \]

**Final Answer**: \((-9)^3 + (4)^3 + (5)^3 = -540\)

### (ii) \((-18)^3 + (9)^3 + (9)^3\)

1. **Identify \(a\), \(b\), and \(c\)**:
- \(a = -18\), \(b = 9\), \(c = 9\)

2. **Calculate \(a + b + c\)**:
\[ a + b + c = -18 + 9 + 9 = 0 \]

3. **Since \(a + b + c = 0\)**:
\[ (-18)^3 + (9)^3 + (9)^3 = 3(-18)(9)(9) \]

4. **Calculate \(3abc\)**:
\[ 3(-18)(9)(9) = 3 \times -18 \times 81 = -4374 \]

**Final Answer**: \((-18)^3 + (9)^3 + (9)^3 = -4374\)

### (iii) \((16)^3 + (1)^3 + (-17)^3\)

1. **Identify \(a\), \(b\), and \(c\)**:
- \(a = 16\), \(b = 1\), \(c = -17\)

2. **Calculate \(a + b + c\)**:
\[ a + b + c = 16 + 1 - 17 = 0 \]

3. **Since \(a + b + c = 0\)**:
\[ (16)^3 + (1)^3 + (-17)^3 = 3(16)(1)(-17) \]

4. **Calculate \(3abc\)**:
\[ 3(16)(1)(-17) = 3 \times 16 \times -17 = -816 \]

**Final Answer**: \((16)^3 + (1)^3 + (-17)^3 = -816\)

### (iv) \((8)^3 + (3)^3 + (-11)^3\)

1. **Identify \(a\), \(b\), and \(c\)**:
- \(a = 8\), \(b = 3\), \(c = -11\)

2. **Calculate \(a + b + c\)**:
\[ a + b + c = 8 + 3 - 11 = 0 \]

3. **Since \(a + b + c = 0\)**:
\[ (8)^3 + (3)^3 + (-11)^3 = 3(8)(3)(-11) \]

4. **Calculate \(3abc\)**:
\[ 3(8)(3)(-11) = 3 \times 8 \times -33 = -792 \]

**Final Answer**: \((8)^3 + (3)^3 + (-11)^3 = -792\)

### Summary of Answers:
1. \((-9)^3 + (4)^3 + (5)^3 = -540\)
2. \((-18)^3 + (9)^3 + (9)^3 = -4374\)
3. \((16)^3 + (1)^3 + (-17)^3 = -816\)
4. \((8)^3 + (3)^3 + (-11)^3 = -792\)