Find the value of (101)⁴ – (99)⁴ / (101)² + (99)².

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Points to Remember:

  • The question requires a mathematical solution.
  • The approach is purely factual and analytical, focusing on applying algebraic identities to simplify the expression.
  • The final answer will be a numerical value.

Introduction:

This question asks us to evaluate the expression [(101)⁴ – (99)⁴] / [(101)² + (99)²]. This can be solved using the difference of squares factorization and simplification techniques. The core concept revolves around recognizing and applying algebraic identities to efficiently compute the result without resorting to direct calculation of large numbers.

Body:

1. Applying the Difference of Squares:

The numerator of the expression, (101)⁴ – (99)⁴, can be factored using the difference of squares identity twice: a² – b² = (a + b)(a – b).

First, let a = (101)² and b = (99)². Then:

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(101)⁴ – (99)⁴ = [(101)²]² – [(99)²]² = [(101)² + (99)²][(101)² – (99)²]

Now, we apply the difference of squares again to (101)² – (99)²:

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(101)² – (99)² = (101 + 99)(101 – 99) = (200)(2) = 400

Therefore, the numerator becomes:

(101)⁴ – (99)⁴ = (101)² + (99)²

2. Simplifying the Expression:

Substituting this back into the original expression:

[(101)⁴ – (99)⁴] / [(101)² + (99)²] = {(101)² + (99)²} / [(101)² + (99)²]

Notice that [(101)² + (99)²] cancels out from the numerator and denominator, leaving:

= 400

Conclusion:

The value of [(101)⁴ – (99)⁴] / [(101)² + (99)²] is 400. This was achieved by strategically applying the difference of squares factorization, simplifying the expression, and eliminating common terms. This demonstrates the power of algebraic manipulation in simplifying complex mathematical expressions and avoiding cumbersome direct calculations. The solution highlights the importance of understanding and applying fundamental algebraic identities for efficient problem-solving. This approach emphasizes a clear, concise, and mathematically sound method, promoting accuracy and efficiency in mathematical computations.

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