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Rationalize the denominator: conjugates

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Learn it If the denominator is $a + b\sqrt{c}$, multiply top and bottom by its conjugate $a - b\sqrt{c}$. The bottom becomes $a^2 - b^2c$, a whole number, because $(p + q)(p - q) = p^2 - q^2$.

Example: $\frac{4}{3 - \sqrt{5}}$

  1. Conjugate: $3 + \sqrt{5}$.
  2. Bottom: $3^2 - (\sqrt{5})^2 = 9 - 5 = 4$.
  3. Top: $4(3 + \sqrt{5}) = 12 + 4\sqrt{5}$.
  4. $\frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}$.

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