0\; ,\; b>0\\\\\sqrt{64a^{10}b^{6}}=\sqrt{8^2(a^5)^2(b^3)^2}=8\cdot |a^5|\cdot |b^3|=8a^5b^3\\\\2)\; \; x<0\; ,\sqrt{25a^{16}x^{10}}=5|a|^8\cdot |x|^5=5\cdot a^8\cdot (-x^5)=-5a^8x^5\\\\3)\; \; a>0\; ,\; \; \frac{b}{a}\cdot \sqrt{\frac{a^2}{b^4}}=\frac{b}{a}\cdot \frac{|a|}{|b|^2}=\frac{b}{a}\cdot \frac{a}{b^2}=\frac{1}{b}\\\\4)\; \; x<0\; ,\; \; 4x^2y\sqrt{\frac{x^{10}}{36y^{12}}}=4x^2y\cdot \frac{|x|^5}{6|y|^6}=4x^2y\cdot \frac{(-x)^5}{6y^6}=\\\\=-4x^2\cdot \frac{x^5}{6y^5}=-\frac{2x^7}{3y^5}" alt="1)\; \; a>0\; ,\; b>0\\\\\sqrt{64a^{10}b^{6}}=\sqrt{8^2(a^5)^2(b^3)^2}=8\cdot |a^5|\cdot |b^3|=8a^5b^3\\\\2)\; \; x<0\; ,\sqrt{25a^{16}x^{10}}=5|a|^8\cdot |x|^5=5\cdot a^8\cdot (-x^5)=-5a^8x^5\\\\3)\; \; a>0\; ,\; \; \frac{b}{a}\cdot \sqrt{\frac{a^2}{b^4}}=\frac{b}{a}\cdot \frac{|a|}{|b|^2}=\frac{b}{a}\cdot \frac{a}{b^2}=\frac{1}{b}\\\\4)\; \; x<0\; ,\; \; 4x^2y\sqrt{\frac{x^{10}}{36y^{12}}}=4x^2y\cdot \frac{|x|^5}{6|y|^6}=4x^2y\cdot \frac{(-x)^5}{6y^6}=\\\\=-4x^2\cdot \frac{x^5}{6y^5}=-\frac{2x^7}{3y^5}" align="absmiddle" class="latex-formula">