Evaluate by using suitable identity: `(997)^(3)`.

Answer

To evaluate \( (997)^3 \) using a suitable identity, we can express 997 as \( 1000 - 3 \). We will use the identity for the cube of a binomial:

\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]

Here, let \( a = 1000 \) and \( b = 3 \). Now we can substitute these values into the identity.

### Step 1: Calculate \( a^3 \)
\[ a^3 = (1000)^3 = 1000000000 \]

### Step 2: Calculate \( b^3 \)
\[ b^3 = (3)^3 = 27 \]

### Step 3: Calculate \( 3a^2b \)
\[ 3a^2b = 3 \times (1000)^2 \times 3 = 3 \times 1000000 \times 3 = 9000000 \]

### Step 4: Calculate \( 3ab^2 \)
\[ 3ab^2 = 3 \times 1000 \times (3)^2 = 3 \times 1000 \times 9 = 27000 \]

### Step 5: Substitute into the identity
Now, substituting all the calculated values into the identity:
\[ (997)^3 = (1000 - 3)^3 = 1000^3 - 3 \times 1000^2 \times 3 + 3 \times 1000 \times 3^2 - 3^3 \]
\[ = 1000000000 - 9000000 + 27000 - 27 \]

### Step 6: Perform the calculations
Now, we will perform the calculations step by step:
1. \( 1000000000 - 9000000 = 991000000 \)
2. \( 991000000 + 27000 = 991027000 \)
3. \( 991027000 - 27 = 991026973 \)

Thus, the final answer is:
\[ (997)^3 = 991026973 \]