A process of removing surds (and/or other complications) from the denominator of a fraction. Related to dividing complex numbers, where we want to remove the imaginary component from the denominator, we use the difference of two squares, multiply by an appropriate fraction, and everything becomes simpler.

Specifically, suppose we want to simplify $\frac{x}{a+\sqrt{d}}.$ We multiply by $\frac{a-\sqrt{d}}{a-\sqrt{d}}$ and simplify:

$\frac{x}{a+\sqrt{d}}\times\frac{a-\sqrt{d}}{a-\sqrt{d}}=\frac{x.(a-\sqrt{d})}{(a+\sqrt{d})(a-\sqrt{d})}=\frac{x.(a-\sqrt{d})}{a^2-d}$

and now the denominator now longer has a surd.

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Surd (none)

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