Question

$\frac{x}{y}^{a-b}*\frac{y}{z} ^{b-c}* \frac{z}{x} ^{c-a}$

Answer 1

Given Expression :-

$\\ \quad \bullet \quad \sf \bigg(\dfrac{x}{y} \bigg)^{a-b} \times \bigg( \dfrac{y}{z} \bigg)^{b-c} \times \bigg( \dfrac{z}{x} \bigg)^{c-a}$

We know that :-

$\\ \quad \mapsto \quad \boxed{ \frak{ \bigg(\dfrac{a}{b} \bigg) ^{c - d} = \dfrac{ {a}^{c} }{ {b}^{d} } } } \bigstar$

Calculation :-

$\\ \quad \longrightarrow \quad \sf \bigg(\dfrac{x}{y} \bigg)^{a-b} \times \bigg( \dfrac{y}{z} \bigg)^{b-c} \times \bigg( \dfrac{z}{x} \bigg)^{c-a}$

$\\ \quad \longrightarrow \quad \sf \dfrac{ {x}^{a} }{ {y}^{b} } \times \dfrac{ {y}^{b} }{ {z}^{c} } \times \dfrac{ {z}^{c} }{ {x}^{a} }$

$\\ \quad \therefore \quad \boxed{ \frak{1}} \bigstar$

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Answer 2

Given,

$\: \: \: \: \: \: \: \frac{x}{y}^{a-b}*\frac{y}{z} ^{b-c}* \frac{z}{x} ^{c-a}$

Solution:-

☆ As, we know that

$\frac{ {x}^{a - b} }{y} = \frac{x {}^{a} }{y{}^{b}}$

Calculation:-

$\frac{x {}^{a} }{y{}^{b}} \times \frac{ {y}^{b} }{z{}^{c}} \times \frac{ {z}^{c} }{x{} ^{a}}$

By, cutting these all, we get

$1$

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Hope it's helpful