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Most of the problems related to irrational numbers with denominators containing irrational numbers need multiplication by conjugate expressions.. So find the complex conjugate and multiply both numerator and denominator.  The n  simplify.

Rationalization\ of\ (denominator)\ fractions\ with\ surds.\\\\\frac{\sqrt{21}}{\sqrt3+\sqrt7}+\frac{2\sqrt5}{\sqrt21+\sqrt5}\\\\=\frac{\sqrt{21}(\sqrt7-\sqrt3)}{(\sqrt3+\sqrt7)(\sqrt7-\sqrt3)}+\frac{2\sqrt5(\sqrt{21}-\sqrt5)}{(\sqrt{21}+\sqrt5)(\sqrt{21}-\sqrt5)}\\\\=\frac{\sqrt{3*7*7}-\sqrt{3*7*3}}{7-3}+\frac{2\sqrt{21*5}-2*5}{21-5}\\\\=\frac{7\sqrt{3}}{4}-\frac{3\sqrt{7}}{4}+\frac{2\sqrt{105}}{16}-\frac{10}{16}\\\\=\frac{28\sqrt{3}-12\sqrt{7}+2\sqrt{105}-10}{16}
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