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JAMB 2010 Mathematics · Question 48

Rationalize 23+55−3\frac{2\sqrt{3} + \sqrt{5}}{\sqrt{5} - \sqrt{3}}

Options

  1. A.(3 √ 15 +11) / 2
  2. B.3 √ 15 -11
  3. C.(3 √ 15 -11) / 2
  4. D.3 √ 15 +11

Answer: A. (3 √ 15 +11) / 2

Explanation written with AI and checked against the answer key

Analyze the question: We need to remove the surd from the denominator by multiplying the top and bottom by the conjugate of the denominator. Concept: The conjugate of (√5 − √3) is (√5 + √3). Multiplying them gives a difference of two squares: (√5 − √3)(√5 + √3) = 5 − 3 = 2 Solution: Multiply top and bottom by (√5 + √3): Numerator: (2√3 + √5)(√5 + √3) = 2√3·√5 + 2√3·√3 + √5·√5 + √5·√3 = 2√15 + 2(3) + 5 + √15 = 3√15 + 11 Denominator: 5 − 3 = 2 So the result is: (3√15 + 11) / 2 Check: Numerically, 2√3 + √5 ≈ 3.464 + 2.236 = 5.700 and √5 − √3 ≈ 2.236 − 1.732 = 0.504, so the ratio is ≈ 11.31. Also, (3(3.873) + 11)/2 = (11.619 + 11)/2 ≈ 11.31. ✓ Tip: When the denominator is of the form (a − b) with surds, always multiply by (a + b). Watch your signs: the "+11" comes from adding 6 and 5, and the √15 terms add up (2√15 + √15 = 3√15). Options with a minus sign can be eliminated quickly.

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