Surds and Radicals

Rationalizing a denominator

To simplify a fraction with a surd in the denominator, multiply top and bottom by the conjugate: this uses the difference-of-squares identity $(a-b)(a+b) = a^2-b^2$ to clear the square root from the bottom.

Worked example

If $x = 2+\sqrt{3}$, find $x^2 + \dfrac{1}{x^2}$.

Solution: Rationalize $\frac{1}{x}$ first:

So $x + \frac{1}{x} = (2+\sqrt{3})+(2-\sqrt{3}) = 4$, and squaring both sides:

Worked example: a telescoping sum

Show that $\dfrac{1}{3-\sqrt{8}} - \dfrac{1}{\sqrt{8}-\sqrt{7}} + \dfrac{1}{\sqrt{7}-\sqrt{6}} - \dfrac{1}{\sqrt{6}-\sqrt{5}} + \dfrac{1}{\sqrt{5}-2} = 5$.

Solution: Every denominator here is a difference of consecutive square roots, and each has the form $(a-b)(a+b)=a^2-b^2$, always equal to exactly 1 once rationalized (e.g. $3^2 - (\sqrt{8})^2 = 9-8=1$). Rationalizing each term:

Every surd term cancels with its neighbor, leaving only $3+2=5$.