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Presentation of Data in Tabular Form — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsPresentation of Data in Tabular Form3 Marks
Question
If x is a positive real number and exponents are rational numbers, simplify $\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{c}}\Big)^{\text{b+c-a}}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^{\text{c+a-b}}\times\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^{\text{a+b-c}}.$
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Answer

$\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{c}}\Big)^{\text{b+c-a}}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^{\text{c+a-b}}\times\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^{\text{a+b-c}}$
$=\bigg(\frac{\text{X}^{\text{b}^2+\text{bc}-\text{ab}}}{\text{X}^{\text{bc}+\text{c}^2-\text{ac}}}\bigg)\times\bigg(\frac{\text{X}^{\text{c}^2+\text{ac}-\text{bc}}}{\text{X}^{\text{ac}+\text{a}^2-\text{ab}}}\bigg)\times\bigg(\frac{\text{X}^{\text{a}^2+\text{ab}-\text{ac}}}{\text{X}^{\text{ab}+\text{b}^2-\text{bc}}}\bigg)$
$=\Big(\text{X}^{\text{b}^2+\text{bc}-\text{ab}-\text{bc}-\text{c}^2+\text{ac}}\Big)$
$\Big(\text{X}^{\text{c}^2+\text{ac}-\text{bc}-\text{ac}-\text{a}^2+\text{ab}}\Big)$
$\Big(\text{X}^{\text{a}^2+\text{ab}-\text{ac}-\text{ab}-\text{b}^2+\text{bc}}\Big)$
$=\Big(\text{X}^{\text{b}^2-\text{ab}-\text{c}^2+\text{ac}}\Big)\Big(\text{X}^{\text{c}^2-\text{bc}-\text{a}^2+\text{ab}}\Big)\Big(\text{X}^{\text{a}^2-\text{ac}-\text{b}^2+\text{bc}}\Big)$
$=\text{X}^{\text{b}^2-\text{ab}-\text{c}^2+\text{ac}+\text{c}^2-\text{bc}-\text{a}^2+\text{ab}+\text{a}^2-\text{ac}-\text{b}^2+\text{bc}}$
$=\text{X}^0$
$=1$

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