Simplify: \((1/7+√2) + (1/7-√2) - (14/49-2)\)

Question

Simplify: \((1/7+√2) + (1/7-√2) - (14/49-2)\)

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

Correct Answer

Option A — 0

Detailed Solution & Explanation

The correct answer is 0.

Key Points

• Add the first two fractions by rationalising over a common denominator:
  • (7 − √2) + (7 + √2) = 14 in the numerator
  • (7 + √2)(7 − √2) = 7² − (√2)² = 49 − 2 = 47 in the denominator
• So the first two terms together equal 14/47.
• The third term is 14/(49 − 2) = 14/47, exactly the same value.
• Therefore 14/47 − 14/47 = 0.
• The identity (a + b)(a − b) = a² − b² is what makes the surds vanish — spot the conjugate pair before doing any arithmetic.

Additional Information

• The conjugate identity (a + b)(a − b) = a² − b² is what removes surds from a denominator, and it underlies all rationalisation.
• To rationalise a single fraction, multiply numerator and denominator by the conjugate of the denominator: 1/(7 + √2) becomes (7 − √2)/47.
• Watch for expressions engineered to cancel completely — when a question offers 0 as an option, check for a self-cancelling structure before grinding through the arithmetic.

प्रश्न (हिन्दी में)

सरल करें: \((1/7+√2) + (1/7-√2) - (14/49-2)\)

1. A. 0
2. B. 1
3. C. 2
4. D. \(√2\)