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Add and subtract radical fractions: same denominator

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Learn it Same denominator: add or subtract the numerators first (use brackets!), then rationalize once by multiplying by the conjugate.

Example: $\frac{7 + 4\sqrt{5}}{2 + \sqrt{5}} - \frac{1 + 9\sqrt{5}}{2 + \sqrt{5}}$

  1. Numerators: $(7 + 4\sqrt{5}) - (1 + 9\sqrt{5}) = 6 - 5\sqrt{5}$.
  2. Multiply top and bottom by $2 - \sqrt{5}$. Bottom: $4 - 5 = -1$.
  3. Top: $(6 - 5\sqrt{5})(2 - \sqrt{5}) = 12 - 6\sqrt{5} - 10\sqrt{5} + 25 = 37 - 16\sqrt{5}$.
  4. Divide by $-1$: $-37 + 16\sqrt{5}$.

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