Show that:$ \left(\frac{x^{a^{2}+b^{2}}}{x^{a b}}\right)^{a+b}\left(\frac{x^{b^{2}+c^{2}}}{x^{b c}}\right)^{b+c}\left(\frac{x^{c^{2}+a^{2}}}{x^{a c}}\right)^{a+c}= x^{2\left(a^{3}+b^{3}+c^{3}\right)} $

To do: 

We have to show that \( \left(\frac{x^{a^{2}+b^{2}}}{x^{a b}}\right)^{a+b}\left(\frac{x^{b^{2}+c^{2}}}{x^{b c}}\right)^{b+c}\left(\frac{x^{c^{2}+a^{2}}}{x^{a c}}\right)^{a+c}=  x^{2\left(a^{3}+b^{3}+c^{3}\right)} \)

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

LHS $=(\frac{x^{a^{2}+b^{2}}}{x^{a b}})^{a+b}(\frac{x^{b^{2}+c^{2}}}{x^{b c}})^{b+c}(\frac{x^{c^{2}+a^{2}}}{x^{a c}})^{a+c}$

$=(x^{a^{2}+b^{2}-a b})^{a+b}(x^{b^{2}+c^{2}-b c})^{b+c}(x^{c^{2}+a^{2}-c a})^{a+c}$

$=x^{(a+b)(a^{2}-a b+b^{2})} \times x^{(b+c)(b^{2}-b c+c^{2})}\times x^{(c+a)(a^{2}-a c+c^{2})}$

$=x^{a^{3}+b^{3}+b^{3}+c^{3}+c^{3}+a^{3}}$

$=x^{2(a^{3}+b^{3}+c^{3})}$

$=$ RHS

Hence proved.