Simplify

Solution:

(i) $\left(\frac{x^{a+b}}{x^{c}}\right)^{a-b}\left(\frac{x^{b+c}}{x^{a}}\right)^{b-c}\left(\frac{x^{c+a}}{x^{b}}\right)^{c-a}$

$\left(x^{a+b-c}\right)^{a-b}\left(x^{b+c-a}\right)^{b-c}\left(x^{c+a-b}\right)^{c-a}$

$\left(x^{a^{2}-b^{2}-c a+c b}\right)\left(x^{b^{2}-c^{2}-a b+a c}\right)\left(x^{c^{2}-a^{2}-b c+a b}\right)$

$x^{a^{2}-b^{2}-c a+c b+b^{2}-c^{2}-a b+a c+c^{2}-a^{2}-b c+a b}$

$x^{0}=1$

(ii) $\sqrt[1 m]{\frac{x^{1}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n 1]{\frac{x^{n}}{x^{1}}}$

$=\sqrt[\operatorname{lm}]{\mathrm{x}^{1}-\mathrm{m}} \times \sqrt[m n]{\mathrm{x}^{\mathrm{m}-\mathrm{n}}} \times \sqrt[n 1]{\mathrm{x}^{\mathrm{n}-1}}$

$=\left(x^{1-m}\right)^{\frac{1}{m}} \times\left(x^{m-n}\right)^{\frac{1}{m n}} \times\left(x^{n-1}\right)^{\frac{1}{n l}}$

$=(\mathrm{x})^{\frac{1-\mathrm{m}}{\mathrm{lm}}} \times(\mathrm{x})^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}} \times(\mathrm{x})^{\frac{\mathrm{n}-1}{\mathrm{nl}}}$

$=(\mathrm{x})^{\frac{1-\mathrm{m}}{\mathrm{lm}}+\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}+\frac{\mathrm{n}-1}{\mathrm{nl}}}$

$=(\mathrm{x})^{\mathrm{n}\left(\frac{1-\mathrm{m}}{\operatorname{lm}}\right)+\mathrm{l}\left(\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}\right)+\mathrm{m}\left(\frac{\mathrm{n}-\mathrm{l}}{\mathrm{nl}}\right)}$

$=(\mathrm{x})^{\frac{\mathrm{nl}-\mathrm{mn}+\mathrm{lm}-\mathrm{nl}+\mathrm{mn}-\mathrm{ml}}{\mathrm{mnl}}}$

$=(\mathrm{x})^{\frac{0}{\mathrm{mnl}}}$

$\mathrm{x}^{0}=1$