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JAMB 1997 · UME · Q7

Simplify 23+3535−23\dfrac{2\sqrt3 + 3\sqrt5}{3\sqrt5 - 2\sqrt3}.

Worked solution (try it first)
  1. Multiply the top and bottom by the conjugate of the bottom, 35+233\sqrt5 + 2\sqrt3.
  2. Bottom: (35)2−(23)2=45−12(3\sqrt5)^2 - (2\sqrt3)^2 = 45 - 12, which is 33.
  3. Top: (23+35)2=12+1215+45(2\sqrt3 + 3\sqrt5)^2 = 12 + 12\sqrt{15} + 45, which is 57+121557 + 12\sqrt{15}.
  4. Divide top and bottom by 3: 19+41511\dfrac{19 + 4\sqrt{15}}{11}, option A.

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