0:\; \sqrt{18y^6}=\sqrt{2\cdot 9\cdot (y^3)^2}=3\cdot |y^3|\sqrt2=3\cdot |y|^3\cdot \sqrt2=3y^3\, \sqrt2\\\\y<0:\; \sqrt{18y^6}=3\cdot |y^3|\sqrt2=3\cdot |y|^3\cdot \sqrt2=3\cdot (-y)^3\, \sqrt2=-2y^3\, \sqrt2" alt="1)\; \; 0,2\sqrt{1000}=0,2\sqrt{100\cdot 10}=0,2\cdot 10\sqrt{10}=2\sqrt{10}\\\\-\frac{1}{8}\sqrt{192}=-\frac{1}{8}\sqrt{64\cdot 3}=-\frac{1}{8}\cdot 8\sqrt3=-\sqrt3\\\\\boxed {\sqrt{x^2}=|x|=\left \{ {{x\; ,\; esli\; x\geq 0} \atop {-x\; ,\; esli\; x<0}} \right. }\\\\\\y>0:\; \sqrt{18y^6}=\sqrt{2\cdot 9\cdot (y^3)^2}=3\cdot |y^3|\sqrt2=3\cdot |y|^3\cdot \sqrt2=3y^3\, \sqrt2\\\\y<0:\; \sqrt{18y^6}=3\cdot |y^3|\sqrt2=3\cdot |y|^3\cdot \sqrt2=3\cdot (-y)^3\, \sqrt2=-2y^3\, \sqrt2" align="absmiddle" class="latex-formula">
0:\; 2c\sqrt7=\sqrt{(2c)^2\cdot 7}=\sqrt{28c^2}\\\\d<0:\; 5d\sqrt2=-\sqrt{(5d)^2\cdot 2}=-\sqrt{50d^2}\\\\3)\; \; a)\; \; (\sqrt5+\sqrt7)^2\; \vee\; (\sqrt2+\sqrt{10})^2\\\\12+2\sqrt{35}\; \vee \; 12+2\sqrt{20}\\\\\sqrt{35}>\sqrt{20}\; \; \Rightarrow \; \; (\sqrt5+\sqrt7)>(\sqrt2+\sqrt{10})\\\\b)\; \; -\frac{1}{7}\sqrt{98}\; \vee \; -2\sqrt2\\\\-\frac{1}{7}\sqrt{49\cdot 2}\; \vee \; -2\sqrt2" alt="2)\; \; 2\sqrt{3a}=\sqrt{2^2\cdot 3a}=\sqrt{12a}\\\\-0,1\sqrt{2b}=-\sqrt{0,1^2\cdot 2b}=-\sqrt{0,02b}\\\\c>0:\; 2c\sqrt7=\sqrt{(2c)^2\cdot 7}=\sqrt{28c^2}\\\\d<0:\; 5d\sqrt2=-\sqrt{(5d)^2\cdot 2}=-\sqrt{50d^2}\\\\3)\; \; a)\; \; (\sqrt5+\sqrt7)^2\; \vee\; (\sqrt2+\sqrt{10})^2\\\\12+2\sqrt{35}\; \vee \; 12+2\sqrt{20}\\\\\sqrt{35}>\sqrt{20}\; \; \Rightarrow \; \; (\sqrt5+\sqrt7)>(\sqrt2+\sqrt{10})\\\\b)\; \; -\frac{1}{7}\sqrt{98}\; \vee \; -2\sqrt2\\\\-\frac{1}{7}\sqrt{49\cdot 2}\; \vee \; -2\sqrt2" align="absmiddle" class="latex-formula">
-\sqrt8\; \; \Rightarrow \; \; -\sqrt2>-2\sqrt2\; \; \Rightarrow \\\\-\frac{1}{7}\sqrt{98}>-2\sqrt2" alt="-\frac{1}{7}\cdot 7\sqrt2\; \vee \; -2\sqrt2\\\\-\sqrt2\; \vee \; -2\sqrt2\\\\(-\sqrt2)^2\; \vee \; (-2\sqrt2)^2\\\\2\; \vee\; 8\\\\2<8\; \; \Rightarrow \; \; \sqrt2<\sqrt8\; \; \Rightarrow \; \; -\sqrt2>-\sqrt8\; \; \Rightarrow \; \; -\sqrt2>-2\sqrt2\; \; \Rightarrow \\\\-\frac{1}{7}\sqrt{98}>-2\sqrt2" align="absmiddle" class="latex-formula">