1) $\sqrt{4}$
2) $\frac{\sqrt{65}}{4\sqrt{39}} = \frac{\sqrt{65}}{4\sqrt{39}} \cdot \frac{\sqrt{39}}{\sqrt{39}} = \frac{\sqrt{2535}}{156} = \frac{\sqrt{169 \cdot 15}}{156} = \frac{13\sqrt{15}}{156} = \frac{\sqrt{15}}{12}$
3) $\frac{\sqrt{18}}{3\sqrt{90}} = \frac{\sqrt{18}}{3\sqrt{90}} \cdot \frac{\sqrt{90}}{\sqrt{90}} = \frac{\sqrt{1620}}{3 \cdot 90} = \frac{\sqrt{324 \cdot 5}}{270} = \frac{18\sqrt{5}}{270} = \frac{\sqrt{5}}{15}$
4) $\frac{4}{4g^2 + 5\sqrt{g}}$ (cannot be simplified further without additional context or assumptions)
5) $\frac{\sqrt{5}}{4 + \sqrt{15}} \cdot \frac{4 - \sqrt{15}}{4 - \sqrt{15}} = \frac{\sqrt{5}(4 - \sqrt{15})}{16 - 15} = 4\sqrt{5} - \sqrt{75} = 4\sqrt{5} - 5\sqrt{3}$
6) $\frac{\sqrt{6}}{7\sqrt{15}} = \frac{\sqrt{6}}{7\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}} = \frac{\sqrt{90}}{7 \cdot 15} = \frac{\sqrt{9 \cdot 10}}{105} = \frac{3\sqrt{10}}{105} = \frac{\sqrt{10}}{35}$
7) $\frac{7\sqrt{3} + \sqrt{15}}{7\sqrt{7}} = \frac{7\sqrt{3}}{7\sqrt{7}} + \frac{\sqrt{15}}{7\sqrt{7}} = \frac{\sqrt{3}}{\sqrt{7}} + \frac{\sqrt{15}}{7\sqrt{7}} = \frac{\sqrt{21}}{7} + \frac{\sqrt{105}}{49}$
8) $\frac{7}{3 - \sqrt{4r^2}} = \frac{7}{3 - 2r}$ (assuming $r \geq 0$)
9) $\frac{4\sqrt{39}}{5\sqrt{195}} = \frac{4\sqrt{39}}{5\sqrt{195}} \cdot \frac{\sqrt{195}}{\sqrt{195}} = \frac{4\sqrt{7605}}{5 \cdot 195} = \frac{4\sqrt{1521 \cdot 5}}{975} = \frac{4 \cdot 39\sqrt{5}}{975} = \frac{156\sqrt{5}}{975} = \frac{4\sqrt{5}}{25}$
10) $\frac{3\sqrt{6} - \sqrt{5}}{4\sqrt{15}} = \frac{3\sqrt{6}}{4\sqrt{15}} - \frac{\sqrt{5}}{4\sqrt{15}} = \frac{3\sqrt{90}}{60} - \frac{\sqrt{75}}{60} = \frac{3 \cdot 3\sqrt{10}}{60} - \frac{5\sqrt{3}}{60} = \frac{9\sqrt{10}}{60} - \frac{5\sqrt{3}}{60} = \frac{3\sqrt{10}}{20} - \frac{\sqrt{3}}{12}$
11) $\frac{4 + \sqrt{11}}{4 - 6\sqrt{2}} \cdot \frac{4 + 6\sqrt{2}}{4 + 6\sqrt{2}} = \frac{(4 + \sqrt{11})(4 + 6\sqrt{2})}{16 - 72} = \frac{16 + 24\sqrt{2} + 4\sqrt{11} + 6\sqrt{22}}{-56} = -\frac{16 + 24\sqrt{2} + 4\sqrt{11} + 6\sqrt{22}}{56} = -\frac{4 + 6\sqrt{2} + \sqrt{11} + \frac{3}{2}\sqrt{22}}{14}$
12) $\frac{\sqrt{16}}{\sqrt{625}} = \frac{4}{25}$
Parent Tip: Review the logic above to help your child master the concept of simplifying radical expressions worksheet algebra 2.
