Evaluate the expression (854³ − 276³) / (854² + 854 × 276 + 276²). What is the value of this expression?

Introduction / Context:
This question involves simplifying an algebraic expression with large numbers. Directly computing 854³ and 276³ would be time consuming, but the structure of the expression matches a standard algebraic identity. Recognizing and applying this identity allows us to evaluate the expression quickly and accurately without heavy computation.

Given Data / Assumptions:
We must evaluate (854³ − 276³) / (854² + 854 × 276 + 276²).All numbers are integers and standard arithmetic operations apply.We should look for patterns that match known factorization formulas.

Concept / Approach:
The expression matches the algebraic identity for the difference of cubes. For any real numbers a and b, a³ − b³ can be factorized as (a − b)(a² + a b + b²). Our numerator is a³ − b³ with a = 854 and b = 276, and the denominator is a² + a b + b². Therefore, the numerator can be written as (a − b) times the denominator, which allows a clean cancellation and leaves only a − b.

Step-by-Step Solution:
Step 1: Identify a = 854 and b = 276 in the expression.Step 2: Recall the identity: a³ − b³ = (a − b)(a² + a b + b²).Step 3: Rewrite the numerator: 854³ − 276³ = (854 − 276)(854² + 854 × 276 + 276²).Step 4: Substitute this factorization into the expression.Step 5: The expression becomes (854 − 276)(854² + 854 × 276 + 276²) / (854² + 854 × 276 + 276²).Step 6: Cancel the common factor (854² + 854 × 276 + 276²) from numerator and denominator.Step 7: The expression simplifies to 854 − 276.Step 8: Compute 854 − 276 = 578.

Verification / Alternative check:
To verify, we can do a quick sanity check. Since 854 is larger than 276, the difference 854 − 276 is positive and equals 578. If we attempted a partial numerical calculation of the original expression with approximate values, we would expect a result in this range. The direct use of the identity is exact and well known, so the simplified result is reliable.

Why Other Options Are Wrong:
The values 546 and 607 might come from subtracting incorrect pairs such as 854 − 308 or from arithmetic errors in subtraction. The option None of these would be correct only if no numeric option matched, but here 578 is fully justified by the algebraic identity. Therefore, the other options do not match the correct simplification.

Common Pitfalls:
Some learners attempt to compute the large cubes directly, which is time consuming and prone to mistakes. Others may misremember the identity and think a³ − b³ factors as (a − b)³ or (a − b)(a − b²), which is incorrect. Remembering the exact form (a − b)(a² + a b + b²) is essential for fast simplification.

Final Answer:
The value of the expression is 578.