Question
Prove that:$\frac{\text{x}^{\text{a}(\text{b}-\text{c})}}{\text{x}^{\text{b}(\text{a}-\text{c})}}\div\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{a}}\Big)^\text{c}=1$
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Answer
$\text{L.H.S}=\frac{\text{x}^{\text{a}(\text{b}-\text{c})}}{\text{x}^{\text{b}(\text{a}-\text{c})}}\div\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{a}}\Big)^\text{c}$$=\frac{\text{X}^{\text{ab}-\text{ac}}}{\text{X}^{\text{ab}-\text{bc}}}\div\frac{\text{X}^\text{bc}}{\text{X}^\text{ac}}$
$=\text{X}^{\text{ab}-\text{ac}-\text{ab}+\text{bc}}\div\text{X}^{\text{bc}-\text{ac}}$
$=\text{X}^{\text{bc}-\text{ac}}\div\text{X}^{\text{bc}-\text{ac}}$
$=1$
$=\text{R.H.S}$
$=\text{X}^{\text{ab}-\text{ac}-\text{ab}+\text{bc}}\div\text{X}^{\text{bc}-\text{ac}}$
$=\text{X}^{\text{bc}-\text{ac}}\div\text{X}^{\text{bc}-\text{ac}}$
$=1$
$=\text{R.H.S}$
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