Question
Prove that: $\frac{(\text{x}^{\text{a}+\text{b}})^2(\text{x}^{\text{b}+\text{c}})^2(\text{x}^{\text{c}+\text{a}})^2}{(\text{x}^\text{a}\text{x}^\text{b}\text{x}^\text{c})}=1$
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Answer
$\text{L.H.S}=\frac{(\text{x}^{\text{a}+\text{b}})^2(\text{x}^{\text{b}+\text{c}})^2(\text{x}^{\text{c}+\text{a}})^2}{(\text{x}^\text{a}\text{x}^\text{b}\text{x}^\text{c})}=1$
$=\frac{(\text{X}^{2\text{a}+2\text{b}})(\text{X}^{2\text{b}+2\text{c}})(\text{X}^{2\text{c}+2\text{a}})}{\text{X}^{4\text{a}}\text{X}^{4\text{b}}\text{X}^{4\text{c}}}$
$=\frac{\text{X}^{2\text{a}+2\text{b}+2\text{b}+2\text{c}+2\text{c}+2\text{a}}}{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}$
$=\frac{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}$
$=1$
$=\text{R.H.S}$
$=\frac{(\text{X}^{2\text{a}+2\text{b}})(\text{X}^{2\text{b}+2\text{c}})(\text{X}^{2\text{c}+2\text{a}})}{\text{X}^{4\text{a}}\text{X}^{4\text{b}}\text{X}^{4\text{c}}}$
$=\frac{\text{X}^{2\text{a}+2\text{b}+2\text{b}+2\text{c}+2\text{c}+2\text{a}}}{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}$
$=\frac{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}{\text{X}^{4\text{a}+4\text{b}+4\text{c}}}$
$=1$
$=\text{R.H.S}$
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