Evaluate the following using suitable identities :
(i) `(99)^(3)`
(ii) `(102)^(3)`
(iii) `(998)^(3)`

Answer

To evaluate the cubes of the numbers 99, 102, and 998 using suitable identities, we will use the following identities:

1. **Identity for (a + b)³**:
\[ (a + b)³ = a³ + b³ + 3ab(a + b) \]

2. **Identity for (a - b)³**:
\[ (a - b)³ = a³ - b³ - 3ab(a - b) \]

Now, let's solve each part step by step.

### (i) Evaluate \(99^3\)

1. Rewrite \(99\) as \(100 - 1\):
\[ 99^3 = (100 - 1)^3 \]
Here, \(a = 100\) and \(b = 1\).

2. Apply the identity for \((a - b)^3\):
\[ (100 - 1)^3 = 100^3 - 1^3 - 3 \cdot 100 \cdot 1 \cdot (100 - 1) \]

3. Calculate each term:
- \(100^3 = 1000000\)
- \(1^3 = 1\)
- \(3 \cdot 100 \cdot 1 = 300\)
- \(100 - 1 = 99\)
- Therefore, \(3 \cdot 100 \cdot 1 \cdot 99 = 29700\)

4. Substitute back into the equation:
\[ 99^3 = 1000000 - 1 - 29700 \]
\[ = 1000000 - 1 - 29700 = 970299 \]

### (ii) Evaluate \(102^3\)

1. Rewrite \(102\) as \(100 + 2\):
\[ 102^3 = (100 + 2)^3 \]
Here, \(a = 100\) and \(b = 2\).

2. Apply the identity for \((a + b)^3\):
\[ (100 + 2)^3 = 100^3 + 2^3 + 3 \cdot 100 \cdot 2 \cdot (100 + 2) \]

3. Calculate each term:
- \(100^3 = 1000000\)
- \(2^3 = 8\)
- \(3 \cdot 100 \cdot 2 = 600\)
- \(100 + 2 = 102\)
- Therefore, \(3 \cdot 100 \cdot 2 \cdot 102 = 61200\)

4. Substitute back into the equation:
\[ 102^3 = 1000000 + 8 + 61200 \]
\[ = 1000000 + 8 + 61200 = 1061208 \]

### (iii) Evaluate \(998^3\)

1. Rewrite \(998\) as \(1000 - 2\):
\[ 998^3 = (1000 - 2)^3 \]
Here, \(a = 1000\) and \(b = 2\).

2. Apply the identity for \((a - b)^3\):
\[ (1000 - 2)^3 = 1000^3 - 2^3 - 3 \cdot 1000 \cdot 2 \cdot (1000 - 2) \]

3. Calculate each term:
- \(1000^3 = 1000000000\)
- \(2^3 = 8\)
- \(3 \cdot 1000 \cdot 2 = 6000\)
- \(1000 - 2 = 998\)
- Therefore, \(3 \cdot 1000 \cdot 2 \cdot 998 = 11976000\)

4. Substitute back into the equation:
\[ 998^3 = 1000000000 - 8 - 11976000 \]
\[ = 1000000000 - 8 - 11976000 = 994011992 \]

### Final Answers
- \(99^3 = 970299\)
- \(102^3 = 1061208\)
- \(998^3 = 994011992\)