Question

What should replace the question mark ?
\(\frac{854\times 854 × 854-276 × 276 × 276}{854 \times854 + 854×276+276 × 276}=?\)

• A. 1130
• B. 578
• C. 565
• D. 1156

Answer 1

The Correct Option is B

To solve the given expression, observe that the structure of the numerator and the denominator follows the form of a difference and a sum of cubes, respectively. The expression is:

\[\frac{854\times 854\times 854 - 276\times 276\times 276}{854\times854 + 854\times276 + 276\times276}\]

We can rewrite this in terms of cube formulas. Let \(a = 854\) and \(b = 276\). Then the expression becomes:

\[\frac{a^3 - b^3}{a^2 + ab + b^2}\]

Recall the algebraic identity for the difference of cubes:

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

Notice that the denominator exactly matches the second factor of the identity:

\[a^2 + ab + b^2\]

Thus, the entire fraction simplifies by canceling the common terms in the numerator and the denominator:

\[\frac{(a-b)(a^2 + ab + b^2)}{a^2 + ab + b^2} = a-b\]

Now substitute back the values of \(a\) and \(b\):

\[a-b = 854 - 276\]

\[a-b = 578\]

Therefore, the value that should replace the question mark is:

578