Answer:
(i) \[(5{{x}^{2}}-6x)\div 3x\] \[(5{{x}^{2}}-6x)\div 3x\] \[=\frac{5{{x}^{2}}-6x}{3x}\] \[=\frac{5{{x}^{2}}}{3x}-\frac{6x}{3x}=\frac{5}{3}x-2\] \[=\frac{1}{3}(5x-6)\] (ii) \[(3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}})\div {{y}^{4}}\] \[(3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}})\div {{y}^{4}}\] \[=\frac{3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}}}{{{y}^{4}}}\] \[=\frac{3{{y}^{8}}}{{{y}^{4}}}\,-\frac{4{{y}^{6}}}{{{y}^{4}}}+\frac{5{{y}^{4}}}{{{y}^{4}}}\] \[=3{{y}^{4}}-4{{y}^{2}}+5\] (iii) \[8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})\div 4{{x}^{2}}{{y}^{2}}{{z}^{2}}\]\[8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})\div 4{{x}^{2}}{{y}^{2}}{{z}^{2}}\] \[=\frac{8({{x}^{3}}{{y}^{2}}{{z}^{3}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{z}^{2}}{{y}^{2}}{{z}^{3}})}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}\] \[=\frac{8{{x}^{2}}{{y}^{2}}{{z}^{2}}\,(x+y+z)}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}\] \[=\frac{8{{x}^{2}}{{y}^{2}}{{z}^{2}}\,(x+y+z)}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}\] (iv) \[({{x}^{3}}+2{{x}^{2}}+3x)\,\div 2x\] \[({{x}^{3}}+2{{x}^{2}}+3x)\,\div 2x\] \[=\frac{{{x}^{3}}+2{{x}^{2}}+3x}{2x}\] \[=\frac{1}{2}\,({{x}^{2}}+2x+3)\] (v) \[({{p}^{3}}{{q}^{6}}-{{p}^{6}}{{q}^{3}})\div \,{{p}^{3}}{{q}^{3}}\] \[({{p}^{3}}{{q}^{6}}-{{p}^{6}}{{q}^{3}})\div \,{{p}^{3}}{{q}^{3}}\] \[=\frac{{{p}^{3}}{{q}^{6}}-{{p}^{6}}{{q}^{3}}}{{{p}^{3}}{{q}^{3}}}\] \[=\frac{{{p}^{3}}{{q}^{3}}({{q}^{3}}-{{q}^{3}})}{{{p}^{3}}{{q}^{3}}}\] \[={{q}^{3}}-{{p}^{3}}\].
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