Without using a calculator, find the value of: \( \begin{array}{ll}\text { a) } \frac{\sqrt{75}+\sqrt{48}}{\sqrt{12}} & \text { b) } \sqrt{3}(\sqrt{48}-3 \sqrt{75}+2 \sqrt{108} \\ \text { a) } \sqrt{50}+\sqrt{32}-\sqrt{18} & \text { d) } \sqrt[3]{\sqrt{64}}\end{array} \)
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To solve these expressions without using a calculator, let's simplify each term step by step: ### Part a: \[ \frac{\sqrt{75} + \sqrt{48}}{\sqrt{12}} \] First, we simplify each square root: \[ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} \] \[ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \] \[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \] Now substituting back into the expression: \[ \frac{5\sqrt{3} + 4\sqrt{3}}{2\sqrt{3}} = \frac{(5+4)\sqrt{3}}{2\sqrt{3}} = \frac{9\sqrt{3}}{2\sqrt{3}} = \frac{9}{2} \] ### Part b: \[ \sqrt{3}(\sqrt{48} - 3\sqrt{75} + 2\sqrt{108}) \] We already have \(\sqrt{48}\) and \(\sqrt{75}\) simplified previously, so let’s simplify \(\sqrt{108}\): \[ \sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3} \] Now inserting back into the expression: \[ \sqrt{3}(\sqrt{48} - 3\sqrt{75} + 2\sqrt{108}) = \sqrt{3}(4\sqrt{3} - 3(5\sqrt{3}) + 2(6\sqrt{3})) \] Calculating the terms inside: \[ = \sqrt{3}(4\sqrt{3} - 15\sqrt{3} + 12\sqrt{3}) = \sqrt{3}((4 - 15 + 12)\sqrt{3}) = \sqrt{3}(1\sqrt{3}) = \sqrt{3} \times \sqrt{3} = 3 \] ### Part c: \[ \sqrt{50} + \sqrt{32} - \sqrt{18} \] Simplifying each square root: \[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \] \[ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \] \[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \] Putting it all together: \[ 5\sqrt{2} + 4\sqrt{2} - 3\sqrt{2} = (5 + 4 - 3)\sqrt{2} = 6\sqrt{2} \] ### Part d: \[ \sqrt[3]{\sqrt{64}} = \sqrt[3]{8} = 2 \] since \(\sqrt{64} = 8\). ### Final Answers: a) \( \frac{9}{2} \) b) \( 3 \) c) \( 6\sqrt{2} \) d) \( 2 \)
