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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.
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