What is the value of 997^3 (the cube of 997)?

Introduction / Context:
This question asks for the cube of 997, a number close to 1000. Instead of multiplying 997 three times directly, we can use the algebraic identity for the cube of a difference to simplify the calculation. This technique is very useful for handling numbers close to powers of 10.

Given Data / Assumptions:

• We must compute 997^3.
• Options: 991026973, 991029673, 991029773, 991097273.
• All calculations use standard arithmetic rules.

Concept / Approach:
Notice that 997 = 1000 - 3. We can apply the identity (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 with a = 1000 and b = 3. This leverages simple powers like 1000^3 and small cubes like 3^3 for an easier computation.

Step-by-Step Solution:
Step 1: Write 997^3 as (1000 - 3)^3.Step 2: Use (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 with a = 1000, b = 3.Step 3: Compute a^3 = 1000^3 = 1000000000.Step 4: Compute 3a^2b = 3 * 1000^2 * 3 = 3 * 1000000 * 3 = 9000000.Step 5: Compute 3ab^2 = 3 * 1000 * 3^2 = 3 * 1000 * 9 = 27000.Step 6: Compute b^3 = 3^3 = 27.Step 7: Substitute into the identity: 997^3 = 1000000000 - 9000000 + 27000 - 27.Step 8: First subtract 9000000: 1000000000 - 9000000 = 991000000.Step 9: Add 27000: 991000000 + 27000 = 991027000.Step 10: Subtract 27: 991027000 - 27 = 991026973.Step 11: Therefore, 997^3 = 991026973.

Verification / Alternative check:
We can check the last digits. Since 7^3 ends in 343, any cube of a number ending in 7 will end with 343 when considering only the last three digits. Here, 991026973 ends with 973, which matches the pattern from our binomial expansion and aligns with the other terms. The detailed identity based computation is the most reliable method in this context.

Why Other Options Are Wrong:

• 991029673 and 991029773: These are close to the correct answer but involve incorrect middle digits, usually from mis adding 27000 or mis subtracting 27.
• 991097273: This has a different pattern in the hundred thousands and millions place, indicating an error in the 3a^2b term.

Common Pitfalls:
Typical mistakes include sign errors in the formula for (a - b)^3, mis computing 3a^2b or 3ab^2, and mis handling the subtraction of b^3. Writing each contribution separately and using place value carefully helps prevent such errors.

Final Answer:
The value of 997^3 is 991026973.