JAMB 2015 Mathematics · Question 60

Rationalize 5 ÷ (2 − √3)

Options

A.3(2 + √5)
B.2(3 + √5)
C.5(2 + √3)
D.3 √2 + 1

Answer: C. 5(2 + √3)

Explanation written with AI and checked against the answer key

Analyze the question: We need to rationalize the denominator of 5/(2 − √3), which means removing the surd from the bottom of the fraction. Concept: Multiply the top and bottom by the conjugate of the denominator. The conjugate of (2 − √3) is (2 + √3). This works because of the difference of two squares: (a − b)(a + b) = a² − b² Solution: 5/(2 − √3) = 5/(2 − √3) × (2 + √3)/(2 + √3) Numerator: 5(2 + √3) Denominator: (2 − √3)(2 + √3) = 2² − (√3)² = 4 − 3 = 1 So: 5/(2 − √3) = 5(2 + √3)/1 = 5(2 + √3) Check: √3 ≈ 1.732, so 2 − √3 ≈ 0.268 and 5 ÷ 0.268 ≈ 18.66. Also, 5(2 + 1.732) = 5 × 3.732 = 18.66. ✓ Tip: When you see a denominator like (a − √b), multiply by (a + √b) (and vice versa). The denominator will always become a² − b, a rational number. If a² − b happens to be 1, as here, the answer is simply the numerator.

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