Evaluate the expression: \((1/3-√8) - (1/√8-√7) + (1/√7-√6) - (1/√6-√5) + (1/√5-2) = \,?\)

Question

Evaluate the expression:

\((1/3-√8) - (1/√8-√7) + (1/√7-√6) - (1/√6-√5) + (1/√5-2) = \,?\)

1. A. 5 (Correct answer)
2. B. 3
3. C. 2
4. D. 0

Correct Answer

Option A — 5

Detailed Solution & Explanation

The correct answer is 5.

Key Points

• Rationalise each term by multiplying by its conjugate. Because each denominator is a difference whose squares differ by 1, every reciprocal simplifies to a sum:
  • 1/(3 − √8) = 3 + √8, since 9 − 8 = 1
  • 1/(√8 − √7) = √8 + √7, and so on down the chain
  • 1/(√5 − 2) = √5 + 2, since 5 − 4 = 1
• Substituting with the given alternating signs:
  • (3 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + 2)
• Every surd cancels against its neighbour, leaving 3 + 2 = 5.

Additional Information

• This is a telescoping series: the alternating signs are chosen so that each irrational term is destroyed by the next, and only the first and last rational parts survive.
• The identity doing the work is (a + b)(a − b) = a² − b², which turns each reciprocal into its conjugate whenever the difference of squares is 1.
• When a question presents a long chain of surd fractions with alternating signs, expect cancellation — evaluating the terms numerically wastes the time the structure is designed to save.

प्रश्न (हिन्दी में)

अभिव्यक्ति का मूल्यांकन करें:

\((1/3-√8) - (1/√8-√7) + (1/√7-√6) - (1/√6-√5) + (1/√5-2) = \,?\)

1. A. 5
2. B. 3
3. C. 2
4. D. 0