Evaluate 31 × 31 × 31 (that is, 31³).

Introduction / Context:
This question is a straightforward computation of a cube of an integer. You are asked to evaluate 31 multiplied by itself three times, which is written as 31³. Calculating cubes accurately is useful in many algebra and arithmetic problems, and it also helps build confidence with mental or written multiplication techniques.

Given Data / Assumptions:

• We need to compute 31 × 31 × 31.
• Standard integer arithmetic applies with no approximations.
• The options given are close in value, so accurate calculation is important.

Concept / Approach:
To compute 31³, it is efficient to first find 31² and then multiply the result by 31 again. Squaring 31 can be done using the identity (30 + 1)² = 30² + 2 * 30 * 1 + 1². Once 31² is obtained, a second multiplication by 31 yields the cube. Writing down each step clearly reduces the chance of arithmetic slips, especially when dealing with similar looking numbers in the answer choices.

Step-by-Step Solution:
First compute 31² = 31 × 31. Use (30 + 1)² = 30² + 2 * 30 * 1 + 1² = 900 + 60 + 1 = 961. So 31² = 961. Now compute 31³ = 31 × 31² = 31 × 961. Multiply 961 by 31: 961 × 30 = 28830 and 961 × 1 = 961. Add these: 28830 + 961 = 29791. Therefore, 31³ = 29791.

Verification / Alternative check:
As a quick check, we know that 30³ = 27000 and 32³ = 32768. Since 31 lies between 30 and 32, 31³ should be between 27000 and 32768, which 29791 is. Additionally, we can multiply in a different order, say 31 × 31 = 961 and then 961 × 31 again with a calculator or independent calculation. Getting the same result 29791 from two routes confirms that the computation is correct.

Why Other Options Are Wrong:

• 27991 and 29719: These are close in magnitude but result from incorrect intermediate additions or multiplications.
• 29971: Another near miss that can arise from reversing digits or minor arithmetic mistakes.
• 31000: This is a rounded guess and not the exact cube.

Common Pitfalls:
Students sometimes misplace digits when adding partial products, especially with similar numbers like 28830 and 961. Others confuse squaring and cubing, computing 31² instead of 31³. A reliable method is to break the multiplication into two stages, double check the square, then multiply that result by 31 with careful alignment of place values. Taking a moment to estimate the range of the answer also helps catch obviously wrong results.

Final Answer:
The value of 31 × 31 × 31 is 29791.