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JAMB 1991 · UME · Q10

Rationalize 23+3232−23\dfrac{2\sqrt3 + 3\sqrt2}{3\sqrt2 - 2\sqrt3}.

Worked solution (try it first)
  1. Multiply the top and bottom by the conjugate of the bottom, 32+233\sqrt2 + 2\sqrt3.
  2. Bottom: (32)2−(23)2=18−12(3\sqrt2)^2 - (2\sqrt3)^2 = 18 - 12, which is 6.
  3. Top: (23+32)2=12+126+18(2\sqrt3 + 3\sqrt2)^2 = 12 + 12\sqrt6 + 18, which is 30+12630 + 12\sqrt6.
  4. Divide by 6: 5+265 + 2\sqrt6, option B.

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