Question

Simplify:

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

(ii) $lm\sqrt{\frac{x^l}{x^m}}\times mn\sqrt{\frac{x^m}{x^n}}\times nl\sqrt{\frac{x^n}{x^l}}$

Answer 1

(i)
$$\begin{gathered} {\left(\frac{x^{a+b}}{x^c}\right)}^{a-b}{\left(\frac{x^{b+c}}{x^a}\right)}^{b-c}{\left(\frac{x^{c+a}}{x^b}\right)}^{c-a} \\ =\left(\frac{x^{\left(a+b\right)\left(a-b\right)}}{x^{c\left(a-b\right)}}\right)\left(\frac{x^{\left(b+c\right)\left(b-c\right)}}{x^{a\left(b-c\right)}}\right)\left(\frac{x^{\left(c+a\right)\left(c-a\right)}}{x^{b\left(c-a\right)}}\right) \\ =\left(\frac{x^{\left(a+b\right)\left(a-b\right)}}{x^{ca-bc}}\right)\left(\frac{x^{\left(b+c\right)\left(b-c\right)}}{x^{ab-ac}}\right)\left(\frac{x^{\left(c+a\right)\left(c-a\right)}}{x^{bc-ab}}\right) \\ =\left(\frac{x^{\left(a^2-b^2\right)}{x}^{\left(b^2-c^2\right)}{x}^{\left(c^2-a^2\right)}}{x^{ca-bc+ab-ac+bc-ab}}\right) \\ =\frac{x^{a^2-b^2+b^2-c^2+c^2-a^2}}{x^{ca-bc+ab-ac+bc-ab}} \\ =\frac{x^0}{x^0} \\ =1 \end{gathered}$$

(ii)
$$\begin{gathered} \sqrt[lm]{\frac{x^l}{x^m}}\times \sqrt[mn]{\frac{x^m}{x^n}}\times \sqrt[nl]{\frac{x^n}{x^l}} \\ ={\left(\frac{x^l}{x^m}\right)}^{\frac{1}{ml}}\times {\left(\frac{x^m}{x^n}\right)}^{\frac{1}{mn}}\times {\left(\frac{x^n}{x^l}\right)}^{\frac{1}{nl}} \\ ={\left(x^{l-m}\right)}^{\frac{1}{ml}}\times {\left(x^{m-n}\right)}^{\frac{1}{mn}}\times {\left(x^{n-l}\right)}^{\frac{1}{nl}} \\ =x^{\frac{l-m}{ml}}\times x^{\frac{m-n}{mn}}\times x^{\frac{n-l}{nl}} \\ =x^{\frac{l-m}{ml}+\frac{m-n}{mn}+\frac{n-l}{nl}} \\ =x^{\frac{ln-mn+lm-nl+nm-lm}{nml}} \\ =x^0 \\ =1 \end{gathered}$$