Chương I - Căn bậc hai. Căn bậc ba

a: ĐKXĐ: x>0

\(E=\dfrac{\sqrt{x}}{x+2\sqrt{x}}:\left(\dfrac{1}{\sqrt{x}}+\dfrac{\sqrt{x}}{\sqrt{x}+2}\right)\)

\(=\dfrac{1}{\sqrt{x}+2}:\dfrac{\sqrt{x}+2+x}{\sqrt{x}\left(\sqrt{x}+2\right)}\)

\(=\dfrac{1}{\sqrt{x}+2}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{x+\sqrt{x}+2}=\dfrac{\sqrt{x}}{x+\sqrt{x}+2}\)

b: E=2/5

=>\(\dfrac{\sqrt{x}}{x+\sqrt{x}+2}=\dfrac{2}{5}\)

=>\(5\sqrt{x}=2x+2\sqrt{x}+4\)

=>\(2x-3\sqrt{x}+4=0\)

=>\(x-\dfrac{3}{2}\cdot\sqrt{x}+2=0\)

=>\(x-2\cdot\sqrt{x}\cdot\dfrac{3}{4}+\dfrac{9}{16}+\dfrac{23}{16}=0\)

=>\(\left(\sqrt{x}-\dfrac{3}{4}\right)^2+\dfrac{23}{16}=0\)(vô lý)

Vậy: \(x\in\varnothing\)

\(\sqrt{\dfrac{4}{9}}:\sqrt{\dfrac{25}{36}}\)

\(=\dfrac{\sqrt{4}}{\sqrt{9}}:\dfrac{\sqrt{25}}{\sqrt{36}}\)

\(=\dfrac{\sqrt{2^2}}{\sqrt{3^2}}:\dfrac{\sqrt{5^2}}{\sqrt{6^2}}\)

\(=\dfrac{2}{3}:\dfrac{5}{6}\)

\(=\dfrac{2}{3}\cdot\dfrac{6}{5}\)

\(=\dfrac{4}{5}\)

\(\dfrac{\sqrt{80}}{\sqrt{5}}\)

\(=\dfrac{\sqrt{5\cdot16}}{\sqrt{5}}\)

\(=\sqrt{\dfrac{5\cdot16}{5}}\)

\(=\sqrt{16}\)

\(=\sqrt{4^2}\)

\(=4\)

\(\sqrt{3}\cdot\sqrt{27}\)

\(=\sqrt{3}\cdot\sqrt{3^3}\)

\(=\sqrt{3\cdot3^3}\)

\(=\sqrt{3^4}\)

\(=\sqrt{9^2}\)

\(=9\)

\(\sqrt{16x}-2\sqrt{36x}+3\sqrt{9x}=2\left(x\ge0\right)\)

\(\Leftrightarrow\sqrt{4^2\cdot x}-2\sqrt{6^2\cdot x}+3\sqrt{3^2x}=2\)

\(\Leftrightarrow4\sqrt{x}-2\cdot6\sqrt{x}+3\cdot3\sqrt{x}=2\)

\(\Leftrightarrow4\sqrt{x}-12\sqrt{x}+9\sqrt{x}=2\)

\(\Leftrightarrow\sqrt{x}=2\)

\(\Leftrightarrow x=2^2\)

\(\Leftrightarrow x=4\left(tm\right)\)

\(3\sqrt{2x}-5\sqrt{8x}+7\sqrt{18x}\left(x\ge0\right)\)

\(=3\sqrt{2x}-5\sqrt{2^2\cdot2x}+7\sqrt{3^2\cdot2x}\)

\(=3\sqrt{2x}-5\cdot2\sqrt{2x}+7\cdot3\sqrt{2x}\)

\(=3\sqrt{2x}-10\sqrt{2x}+21\sqrt{2x}\)

\(=\left(3-10+21\right)\sqrt{2x}\)

\(=14\sqrt{2x}\)

\(2\sqrt{12}-3\sqrt{48}+2\sqrt{75}\)

\(=2\sqrt{2^2\cdot3}-3\sqrt{2^4\cdot3}+2\sqrt{5^2\cdot3}\)

\(=2\cdot2\sqrt{3}-3\cdot2^2\sqrt{3}+2\cdot5\sqrt{3}\)

\(=4\sqrt{3}-3\cdot4\sqrt{3}+10\sqrt{3}\)

\(=4\sqrt{3}-12\sqrt{3}+10\sqrt{3}\)

\(=\left(4-12+10\right)\sqrt{3}\)

\(=2\sqrt{3}\)

\(\left(\dfrac{1}{\sqrt{5}-\sqrt{2}}-\dfrac{1}{\sqrt{5}+\sqrt{2}}+1\right)\cdot\dfrac{1}{\left(\sqrt{2}+1\right)^2}\)

\(=\dfrac{\sqrt{5}+\sqrt{2}-\sqrt{5}+\sqrt{2}+3}{3}\cdot\dfrac{1}{3+2\sqrt{2}}\)

\(=\dfrac{3+2\sqrt{2}}{3+2\sqrt{2}}\cdot\dfrac{1}{3}=\dfrac{1}{3}\)

Đề bài yêu cầu gì vậy bạn?

\(\sqrt{\left(x+1\right)\left(x-3\right)}\) có nghĩa khi:

\(\left(x+1\right)\left(x-3\right)\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+1\ge0\\x-3\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x+1\le0\\x-3\le0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge-1\\x\ge3\end{matrix}\right.\\\left\{{}\begin{matrix}x\le-1\\x\le3\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x\ge3\\x\le-1\end{matrix}\right.\)

Vậy \(\sqrt{\left(x+1\right)\left(x-3\right)}\) xác định khi: \(\left[{}\begin{matrix}x\le-1\\x\ge3\end{matrix}\right.\)