Solution. (a) A single radical: multiply top and bottom by it. \[\frac {3}{\sqrt {5}}=\frac {3}{\sqrt {5}}\times \frac {\sqrt {5}}{\sqrt {5}} =\frac {3\sqrt {5}}{5}.\] Multiplying by \(\frac {\sqrt 5}{\sqrt 5}\) is multiplying by \(1\), so the value has not changed — only its appearance.

(b) Two terms underneath. Multiply by the conjugate, the same expression with the sign between the terms reversed, because \((x-y)(x+y)=x^2-y^2\) removes both radicals at once. \[\frac {2}{\sqrt {7}-\sqrt {3}}\times \frac {\sqrt {7}+\sqrt {3}}{\sqrt {7}+\sqrt {3}} =\frac {2\left (\sqrt {7}+\sqrt {3}\right )}{\left (\sqrt {7}\right )^2-\left (\sqrt {3}\right )^2} =\frac {2\left (\sqrt {7}+\sqrt {3}\right )}{7-3} =\frac {2\left (\sqrt {7}+\sqrt {3}\right )}{4} =\frac {\sqrt {7}+\sqrt {3}}{2}.\]

(c) Same method; the conjugate of \(3-\sqrt {2}\) is \(3+\sqrt {2}\). \[\frac {1+\sqrt {2}}{3-\sqrt {2}}\times \frac {3+\sqrt {2}}{3+\sqrt {2}} =\frac {\left (1+\sqrt {2}\right )\left (3+\sqrt {2}\right )}{9-2}.\] Expanding the top: \[\left (1+\sqrt {2}\right )\left (3+\sqrt {2}\right ) =3+\sqrt {2}+3\sqrt {2}+\left (\sqrt {2}\right )^2 =3+4\sqrt {2}+2=5+4\sqrt {2}.\] \[\therefore \quad \frac {1+\sqrt {2}}{3-\sqrt {2}}=\frac {5+4\sqrt {2}}{7}.\]

Check. \(\frac {1+\sqrt 2}{3-\sqrt 2}\approx 1.5224\) and \(\frac {5+4\sqrt 2}{7}\approx 1.5224\). A quick numerical check like this catches sign errors in the expansion, and takes seconds.