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Number System — Maths STD 9 — Question

Maharashtra BoardEnglish MediumSTD 9MathsNumber System3 Marks
Question
Prove that:$\Big(\frac{1}{\text{x}^{\text{a}-\text{b}}}\Big)^{\frac{1}{\text{a}-\text{c}}}\times\Big(\frac{1}{\text{x}^{\text{b}-\text{c}}}\Big)^{\frac{1}{\text{b}-\text{a}}}\times\Big(\frac{1}{\text{x}^{\text{c}-\text{a}}}\Big)^{\frac{1}{\text{c}-\text{b}}}=1$
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Answer

$\Big(\frac{1}{\text{x}^{\text{a}-\text{b}}}\Big)^{\frac{1}{\text{a}-\text{c}}}\times\Big(\frac{1}{\text{x}^{\text{b}-\text{c}}}\Big)^{\frac{1}{\text{b}-\text{a}}}\times\Big(\frac{1}{\text{x}^{\text{c}-\text{a}}}\Big)^{\frac{1}{\text{c}-\text{b}}}=1$$=\big(\text{x}\big)^{\frac{1}{(\text{a}-\text{b})}\times\frac{1}{(\text{a}-\text{c})}}\times=\big(\text{x}\big)^{\frac{1}{(\text{b}-\text{c})}\times\frac{1}{(\text{b}-\text{a})}}\times\big(\text{x}\big)^{\frac{1}{(\text{c}-\text{a})}\times\frac{1}{(\text{c}-\text{b})}}$
$=\big(\text{x}\big)^{\frac{1}{(\text{a}-\text{b})(\text{a}-\text{c})}+\frac{1}{(\text{b}-\text{c})(\text{b}-\text{a})}+\frac{1}{(\text{c}-\text{a})(\text{c}-\text{b})}}$
$$$$$$$=\big(\text{x}\big)^{\frac{1}{(\text{a}-\text{b})(\text{c}-\text{a})}-\frac{1}{(\text{b}-\text{c})(\text{a}-\text{b})}-\frac{1}{(\text{c}-\text{a})(\text{b}-\text{c})}}$
$=\big(\text{x}\big)^{\frac{-(\text{b}-\text{c})-(\text{c}-\text{a})-(\text{a}-\text{b})}{(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})}}$
$=\big(\text{x}\big)^{\frac{-\text{b}+\text{c}-\text{c}+\text{a}-\text{a}+\text{b}}{(\text{a}-\text{b}(\text{b}-\text{c})(\text{c}-\text{a})}}$
$=\text{x}^0$
$=1$
$=\text{R.H.S}$

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