1) $\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}$
2) $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$
3) $\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$
4) $\sqrt{175} = \sqrt{25 \cdot 7} = \sqrt{25} \cdot \sqrt{7} = 5\sqrt{7}$
5) $\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}$
6) $\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}$
7) $4\sqrt{3} - \sqrt{27} = 4\sqrt{3} - \sqrt{9 \cdot 3} = 4\sqrt{3} - 3\sqrt{3} = (4 - 3)\sqrt{3} = \sqrt{3}$
8) $-10\sqrt{11} - 11\sqrt{11} = (-10 - 11)\sqrt{11} = -21\sqrt{11}$
9) $3\sqrt{5} + 7\sqrt{5} = (3 + 7)\sqrt{5} = 10\sqrt{5}$
10) $-11\sqrt{21} - 11\sqrt{21} = (-11 - 11)\sqrt{21} = -22\sqrt{21}$
11) $-9\sqrt{15} + 10\sqrt{15} = (-9 + 10)\sqrt{15} = \sqrt{15}$
12) $3\sqrt{24} + 3\sqrt{81} = 3\sqrt{4 \cdot 6} + 3 \cdot 9 = 3 \cdot 2\sqrt{6} + 27 = 6\sqrt{6} + 27$
13) $\sqrt{10} \cdot \sqrt{14} = \sqrt{10 \cdot 14} = \sqrt{140} = \sqrt{4 \cdot 35} = \sqrt{4} \cdot \sqrt{35} = 2\sqrt{35}$
14) $\sqrt[3]{2} \cdot \sqrt[3]{4} = \sqrt[3]{2 \cdot 4} = \sqrt[3]{8} = 2$
15) $2\sqrt{14} \cdot 3\sqrt{21} = 2 \cdot 3 \cdot \sqrt{14 \cdot 21} = 6\sqrt{294} = 6\sqrt{49 \cdot 6} = 6 \cdot 7\sqrt{6} = 42\sqrt{6}$
16) $\sqrt[3]{\frac{10}{9}} = \frac{\sqrt[3]{10}}{\sqrt[3]{9}}$ (This is already simplified as the cube root of a fraction cannot be simplified further without rationalizing, but the problem does not require rationalizing the denominator for cube roots in the same way as square roots.)
17) $\frac{1 + \sqrt{2}}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{(1 + \sqrt{2})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{3 - \sqrt{5} + 3\sqrt{2} - \sqrt{10}}{9 - 5} = \frac{3 - \sqrt{5} + 3\sqrt{2} - \sqrt{10}}{4}$
18) $\frac{5}{2 - \sqrt{7}} \cdot \frac{2 + \sqrt{7}}{2 + \sqrt{7}} = \frac{5(2 + \sqrt{7})}{(2 - \sqrt{7})(2 + \sqrt{7})} = \frac{10 + 5\sqrt{7}}{4 - 7} = \frac{10 + 5\sqrt{7}}{-3} = -\frac{10}{3} - \frac{5\sqrt{7}}{3}$
Parent Tip: Review the logic above to help your child master the concept of divide radicals worksheet.
