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a) \(\sqrt{27x^2}=\sqrt{3.\left(3x\right)^2}=\left|3x\right|.\sqrt{3}=3x\sqrt{3}\left(x>0\right)\)
b) \(\sqrt{8xy^2}=\left|y\right|.2\sqrt{2x}=-2y\sqrt{2x}\left(x\ge0,y\le0\right)\)
1) \(x\sqrt{13}=\sqrt{13x^2}\left(x\ge0\right)\)
2) \(x\sqrt{-15x}=-\left|x\right|\sqrt{15x}=-\sqrt{15x^3}\left(x< 0\right)\)
3) \(x\sqrt{2}=-\left|x\right|\sqrt{2}=-\sqrt{2x^2}\left(x\le0\right)\)
3\(\sqrt{5}\)= \(\sqrt{3^2.5}\)=\(\sqrt{45}\)
-5\(\sqrt{2}\)= \(-\sqrt{5^2.2}\)= -\(\sqrt{50}\)
\(\dfrac{-2}{3}\sqrt{xy}\) = \(-\sqrt{\left(\dfrac{2}{3}\right)^2xy}\) = -\(\sqrt{\dfrac{4}{9}xy}\)
x\(\sqrt{\dfrac{2}{x}}\)= \(\sqrt{\dfrac{2x^2}{x}}=\sqrt{2x}\)
\(\frac{1}{2x-1}\sqrt{5-20x+20x^2}=\frac{1}{2x-1}\sqrt{5.\left(1-4x+4x^2\right)}\)
\(=\frac{1}{2x-1}\sqrt{5.\left(1-2x\right)^2}=\sqrt{\frac{1}{\left(2x-1\right)^2}}\sqrt{5.\left(2x-1\right)^2}\)(x>1/2)
\(=\sqrt{\frac{1}{\left(2x-1\right)^2}.5.\left(2x-1\right)^2}=\sqrt{5}\)
\(3\sqrt{5}=\sqrt{45}\)
\(-5\sqrt{2}=-\sqrt{25}.\sqrt{2}=-\sqrt{50}\)
\(\dfrac{-2}{3}\sqrt{xy}=-\sqrt{\dfrac{4}{9}}.\sqrt{xy}=-\sqrt{\dfrac{4}{9}xy}\left(xy\ge0\right)\)
\(x\sqrt{\dfrac{2}{x}}=\sqrt{x^2}.\sqrt{\dfrac{2}{x}}=\sqrt{\dfrac{2x^2}{x}}=\sqrt{2x}\left(x>0\right)\)
a) \(x\sqrt{13}=\sqrt{x^2\cdot13}=\sqrt{13x^2}\)
b) \(x\sqrt{2}=\sqrt{x^2\cdot2}=\sqrt{2x^2}\)
c) \(x\sqrt{\frac{-11}{x}}=\sqrt{x^2\cdot\frac{-11}{x}}=\sqrt{-11x}\)
a, \(\sqrt{\frac{\left(x-y\right)^2}{x^2}\cdot\frac{x}{x-y}}=\) \(\frac{x-y}{x}\)
b. \(\sqrt{\frac{\left(x+y\right)^2}{\left(x-y\right)^2}\cdot\frac{x-y}{x+y}}=\sqrt{\frac{x+y}{x-y}}\)
c.\(\sqrt{\frac{x^4}{\left(x-5\right)^2}\cdot\frac{x-5}{3x}}=\sqrt{\frac{x^3}{3\left(x-5\right)}}\)
\(-2\sqrt{-a}=\sqrt{\left(-2\right)^2\cdot-a}=\sqrt{-4a}\)
\(\left(x-5\right)\sqrt{\frac{3}{25-x^2}}=\sqrt{\left(x-5\right)^2}\sqrt{\frac{3}{\left(5-x\right)\left(x+5\right)}}=\sqrt{\left(5-x\right)^2.\frac{3}{\left(5-x\right)\left(x+5\right)}}=\sqrt{\frac{3\left(5-x\right)}{x+5}}\)
