\(a,\sqrt{3x}.\sqrt{\dfrac{12}{x}}=\sqrt{\dfrac{3x.12}{x}}=\sqrt{36}=\sqrt{6^2}=6\\ b,\dfrac{\sqrt{7y^3}}{\sqrt{63y}}=\sqrt{\dfrac{7}{63}.\dfrac{y^3}{y}}=\sqrt{\dfrac{1}{9}y^2}=\sqrt{\left(\dfrac{1}{3}y\right)^2}=\dfrac{1}{3}y\)
\(c,\sqrt{14a^3}.\sqrt{\dfrac{25b^6}{126a}}=\sqrt{\dfrac{14.25.a^3.b^6}{126a}}\\ =\sqrt{\dfrac{350}{126}a^2b^6}=\sqrt{\dfrac{25}{9}a^2b^6}=\sqrt{\left(\dfrac{5}{3}ab^3\right)^2}=\left|\dfrac{5}{3}ab^3\right|=-\dfrac{5}{3}ab^3\\ d,\left(x-y\right)\sqrt{\dfrac{y^2}{x^2+y^2-2xy}}=\left(x-y\right)\sqrt{\dfrac{y^2}{\left(x-y\right)^2}}=\left(x-y\right).\sqrt{\left(\dfrac{y}{x-y}\right)^2}\\ =\left(x-y\right).\left|\dfrac{y}{x-y}\right|=-\left(x-y\right).\dfrac{y}{x-y}=-y\)
