[tex]\bf \sqrt{14q}\cdot 2\sqrt{4q}\implies \sqrt{14q}\cdot 2\sqrt{2^2q}\implies \sqrt{14q}\cdot 2(2)\sqrt{q}
\\\\\\
\sqrt{14q}\cdot 4\sqrt{q}
\implies
4\sqrt{14q}\cdot \sqrt{q}\implies 4\sqrt{14q\cdot q}\implies 4\sqrt{14q^2}
\\\\\\
4q\sqrt{14}[/tex]
now, for the next one, bear in mind that, if you move a "factor" from the bottom to the top, the exponent changes sign, and if you move it from the top to the bottom, also changes sign, so, let's see,
[tex]\bf \left.\qquad \qquad \right.\textit{negative exponents}\\\\
a^{-{ n}} \implies \cfrac{1}{a^{ n}}
\qquad \qquad
\cfrac{1}{a^{ n}}\implies a^{-{ n}}
\qquad \qquad
a^{{{ n}}}\implies \cfrac{1}{a^{-{{ n}}}}\\\\
-------------------------------\\\\
\sqrt{\cfrac{63x^{15}y^9}{7xy^{11}}}\implies \sqrt{\cfrac{63}{7}\cdot \cfrac{x^{15}y^9}{xy^{11}}}\implies \sqrt{ \cfrac{9x^{15}y^9}{xy^{11}}}\implies \sqrt{ \cfrac{9x^{15}x^{-1}}{y^{11}y^{-9}}}[/tex]
[tex]\bf \sqrt{ \cfrac{9x^{15-1}}{y^{11-9}}}\implies \sqrt{\cfrac{9x^{14}}{y^2}}\implies \sqrt{\cfrac{9x^{7\cdot 2}}{y^2}}\implies \sqrt{\cfrac{3^2(x^7)^2}{y^2}}\implies \cfrac{3x^7}{y}\\\\
-------------------------------[/tex]
[tex]\bf 2\sqrt{6}+3\sqrt{96}\qquad
\begin{cases}
96=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\\
\qquad 2^2\cdot 2^2\cdot 2\cdot 3\\
\qquad (2\cdot 2)^2\cdot 2\cdot 3\\
\qquad 4^2\cdot 6
\end{cases}\implies 2\sqrt{6}+3\sqrt{4^2\cdot 6}
\\\\\\
2\sqrt{6}+3(4)\sqrt{6}\implies 2\sqrt{6}+12\sqrt{6}\implies 14\sqrt{6}[/tex]
