4\; \; -\; otvet" alt="x=7:\; \; 2\sqrt{7-3}=2\sqrt4=2\cdot 2=4\\\\1+\sqrt{7+2}=1+\sqrt9=1+3=4\; ,\; \; \; 4=4\\\\Otvet:\; \; x=7\; .\\\\105.9)\; \; \sqrt{x-4}=\sqrt{x-3}-\sqrt{2x-1}\; ,\; \; ODZ:\; \left \{ {{x-4 \geq 0} \atop {x-3 \geq 0,\; 2x-1 \geq 0}} \right. \; ,\; x \geq 4\\\\x-4=(x-3)-2\sqrt{(x-3)(2x-1)}+(2x-1)\\\\2\sqrt{2x^2-7x+3}=2x\; |:2\; ,\; \; \; \; \sqrt{2x^2-7x+3}=x\\\\2x^2-7x+3=x^2\; \; ,\; \; x^2-7x+3=0\\\\D=49-12=37\; ,\\\\x_1=\frac{7-\sqrt{37}}{4}\approx 0,46\ \textless \ 4\; \; \to \; \; postoronnij\; koren\\\\x_2=\frac{7+\sqrt{37}}{4}\approx 6,54>4\; \; -\; otvet" align="absmiddle" class="latex-formula">
![106.1)\; \; \sqrt{x} -5 \sqrt[4]{x} +6=0\; ,\; \; ODZ:\; x \geq 0\\\\Tak\; kak\; \; (\sqrt[4]{x})^2= \sqrt[4]{x^2} =\sqrt{x} \; ,\; to\; \; t=\sqrt[4]{x}\; \; \Rightarrow \; \; t^2=\sqrt{x}\\\\t^2-5t+6=0\; \; \to \; \; t_1=2,\; t_2=3\; \; (teorema\; Vieta)\\\\\sqrt[4]{x}=2\; \; \to \; \; x=2^4=16\\\\ \sqrt[4]{x}=3\; \; \to \; \; x=3^4=81\\\\Otvet:\; \; x=16\; ,\; x=81\; .](https://tex.z-dn.net/?f=106.1%29%5C%3B+%5C%3B++%5Csqrt%7Bx%7D+-5+%5Csqrt%5B4%5D%7Bx%7D+%2B6%3D0%5C%3B+%2C%5C%3B+%5C%3B+ODZ%3A%5C%3B+x+%5Cgeq+0%5C%5C%5C%5CTak%5C%3B+kak%5C%3B+%5C%3B+%28%5Csqrt%5B4%5D%7Bx%7D%29%5E2%3D+%5Csqrt%5B4%5D%7Bx%5E2%7D+%3D%5Csqrt%7Bx%7D+%5C%3B+%2C%5C%3B+to%5C%3B+%5C%3B+t%3D%5Csqrt%5B4%5D%7Bx%7D%5C%3B+%5C%3B+%5CRightarrow+%5C%3B+%5C%3B+t%5E2%3D%5Csqrt%7Bx%7D%5C%5C%5C%5Ct%5E2-5t%2B6%3D0%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+t_1%3D2%2C%5C%3B+t_2%3D3%5C%3B+%5C%3B+%28teorema%5C%3B+Vieta%29%5C%5C%5C%5C%5Csqrt%5B4%5D%7Bx%7D%3D2%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+x%3D2%5E4%3D16%5C%5C%5C%5C+%5Csqrt%5B4%5D%7Bx%7D%3D3%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+x%3D3%5E4%3D81%5C%5C%5C%5COtvet%3A%5C%3B+%5C%3B+x%3D16%5C%3B+%2C%5C%3B+x%3D81%5C%3B+. "106.1)\; \; \sqrt{x} -5 \sqrt[4]{x} +6=0\; ,\; \; ODZ:\; x \geq 0\\\\Tak\; kak\; \; (\sqrt[4]{x})^2= \sqrt[4]{x^2} =\sqrt{x} \; ,\; to\; \; t=\sqrt[4]{x}\; \; \Rightarrow \; \; t^2=\sqrt{x}\\\\t^2-5t+6=0\; \; \to \; \; t_1=2,\; t_2=3\; \; (teorema\; Vieta)\\\\\sqrt[4]{x}=2\; \; \to \; \; x=2^4=16\\\\ \sqrt[4]{x}=3\; \; \to \; \; x=3^4=81\\\\Otvet:\; \; x=16\; ,\; x=81\; .")
![106.5)\; \; \sqrt[3]{9-6x+x^2}-\sqrt[3]{3-x}-2=0\; ,\; \; ODZ:\; \; x\in R\\\\ \sqrt[3]{(3-x)^2}-\sqrt[3]{3-x} -2=0\\\\t= \sqrt[3]{3-x} \; ,\; \; t^2-t-2=0\\\\t_1=-1\; ,\; \; t_2=2\; \; (teorema\; Vieta)\\\\ \sqrt[3]{3-x}=-1\; \; \to \; \; 3-x=(-1)^3\; ,\; \;3- x=-1\; ,\; \; x=4\\\\\sqrt[3]{3-x}=2\; \; \to \; \; 3-x=8\; ,\; \; x=-5\\\\Otvet:\; \; x=4\; ,\; \; x=-5\; .](https://tex.z-dn.net/?f=106.5%29%5C%3B+%5C%3B+%5Csqrt%5B3%5D%7B9-6x%2Bx%5E2%7D-%5Csqrt%5B3%5D%7B3-x%7D-2%3D0%5C%3B+%2C%5C%3B+%5C%3B+ODZ%3A%5C%3B+%5C%3B+x%5Cin+R%5C%5C%5C%5C+%5Csqrt%5B3%5D%7B%283-x%29%5E2%7D-%5Csqrt%5B3%5D%7B3-x%7D+-2%3D0%5C%5C%5C%5Ct%3D+%5Csqrt%5B3%5D%7B3-x%7D+%5C%3B+%2C%5C%3B+%5C%3B+t%5E2-t-2%3D0%5C%5C%5C%5Ct_1%3D-1%5C%3B+%2C%5C%3B+%5C%3B+t_2%3D2%5C%3B+%5C%3B+%28teorema%5C%3B+Vieta%29%5C%5C%5C%5C+%5Csqrt%5B3%5D%7B3-x%7D%3D-1%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+3-x%3D%28-1%29%5E3%5C%3B+%2C%5C%3B+%5C%3B3-+x%3D-1%5C%3B+%2C%5C%3B+%5C%3B+x%3D4%5C%5C%5C%5C%5Csqrt%5B3%5D%7B3-x%7D%3D2%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+3-x%3D8%5C%3B+%2C%5C%3B+%5C%3B+x%3D-5%5C%5C%5C%5COtvet%3A%5C%3B+%5C%3B+x%3D4%5C%3B+%2C%5C%3B+%5C%3B+x%3D-5%5C%3B+. "106.5)\; \; \sqrt[3]{9-6x+x^2}-\sqrt[3]{3-x}-2=0\; ,\; \; ODZ:\; \; x\in R\\\\ \sqrt[3]{(3-x)^2}-\sqrt[3]{3-x} -2=0\\\\t= \sqrt[3]{3-x} \; ,\; \; t^2-t-2=0\\\\t_1=-1\; ,\; \; t_2=2\; \; (teorema\; Vieta)\\\\ \sqrt[3]{3-x}=-1\; \; \to \; \; 3-x=(-1)^3\; ,\; \;3- x=-1\; ,\; \; x=4\\\\\sqrt[3]{3-x}=2\; \; \to \; \; 3-x=8\; ,\; \; x=-5\\\\Otvet:\; \; x=4\; ,\; \; x=-5\; .")
![106.2)\; \; 3\sqrt[3]{x}+5\sqrt[6]{x}-2=0\; ,\; \; ODZ: \; x \geq 0\\\\ t=\sqrt[6]{x} \; ,\; \; t^2=\sqrt[6]{x^2}= \sqrt[3]{x} \; \; \Rightarrow \; \; \; 3t^2+5t-2=0\\\\D=25+24=49\; ,\; \; t_1= \frac{-5-7}{6}=-2\ ,\; t_2= \frac{1}{3}\\\\ \sqrt[6]{x}\ne -2\; ,\; \; tak\; kak\; \; \sqrt[6]{x} \geq 0 \\\\\sqrt[6]{x} = \frac{1}{3} \; \; \to \; \; x=(\frac{1}{3})^6= \frac{1}{729} \; \; -\; \; otvet](https://tex.z-dn.net/?f=106.2%29%5C%3B+%5C%3B++3%5Csqrt%5B3%5D%7Bx%7D%2B5%5Csqrt%5B6%5D%7Bx%7D-2%3D0%5C%3B+%2C%5C%3B+%5C%3B+ODZ%3A+%5C%3B+x+%5Cgeq+0%5C%5C%5C%5C+t%3D%5Csqrt%5B6%5D%7Bx%7D+%5C%3B+%2C%5C%3B+%5C%3B+t%5E2%3D%5Csqrt%5B6%5D%7Bx%5E2%7D%3D+%5Csqrt%5B3%5D%7Bx%7D+%5C%3B+%5C%3B+%5CRightarrow+%5C%3B+%5C%3B+%5C%3B+3t%5E2%2B5t-2%3D0%5C%5C%5C%5CD%3D25%2B24%3D49%5C%3B+%2C%5C%3B+%5C%3B+t_1%3D+%5Cfrac%7B-5-7%7D%7B6%7D%3D-2%5C+%2C%5C%3B+t_2%3D+%5Cfrac%7B1%7D%7B3%7D%5C%5C%5C%5C+%5Csqrt%5B6%5D%7Bx%7D%5Cne+-2%5C%3B+%2C%5C%3B+%5C%3B+tak%5C%3B+kak%5C%3B+%5C%3B+%5Csqrt%5B6%5D%7Bx%7D+%5Cgeq+0+%5C%5C%5C%5C%5Csqrt%5B6%5D%7Bx%7D+%3D+%5Cfrac%7B1%7D%7B3%7D+%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+x%3D%28%5Cfrac%7B1%7D%7B3%7D%29%5E6%3D+%5Cfrac%7B1%7D%7B729%7D+%5C%3B+%5C%3B+-%5C%3B+%5C%3B+otvet "106.2)\; \; 3\sqrt[3]{x}+5\sqrt[6]{x}-2=0\; ,\; \; ODZ: \; x \geq 0\\\\ t=\sqrt[6]{x} \; ,\; \; t^2=\sqrt[6]{x^2}= \sqrt[3]{x} \; \; \Rightarrow \; \; \; 3t^2+5t-2=0\\\\D=25+24=49\; ,\; \; t_1= \frac{-5-7}{6}=-2\ ,\; t_2= \frac{1}{3}\\\\ \sqrt[6]{x}\ne -2\; ,\; \; tak\; kak\; \; \sqrt[6]{x} \geq 0 \\\\\sqrt[6]{x} = \frac{1}{3} \; \; \to \; \; x=(\frac{1}{3})^6= \frac{1}{729} \; \; -\; \; otvet")
P.S. Во всех примерах сделайте проверку, как в №105.8.