Prove that: (1 x^ a-b )^ 1 a-c (1 x^ b-c )^ 1 b-a (1 x^ c-a )^ 1 c-b =1

(1 x^ a-b )^ 1 a-c (1 x^ b-c )^ 1 b-a (1 x^ c-a )^ 1 c-b =1 = (x )^ 1 (a-b) 1 (a-c) = (x )^ 1 (b-c) 1 (b-a) (x )^ 1 (c-a) 1 (c-b) = (x )^ 1 (a-b)(a-c) +1 (b-c)(b-a) +1 (c-a)(c-b) = (x )^ 1 (a-b)(c-a) -1 (b-c)(a-b) -1 (c-a)(b-c) = (x )^ -(b-c)-(c-a)-(a-b) (a-b)(b-c)(c-a) = (x )^ -b+c-c+a-a+b (a-b(b-c)(c-a) =x^0 =1 =R.H.S

Prove that: (1 x^ a-b )^ 1 a-c (1 x^ b-c )^ 1 b-a (1 x^ c-a )^ 1 …

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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

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