Question :- Simplify the following :
[tex]\sf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } \\ \\ [/tex]
Answer:
[tex]\qquad\boxed{ \sf{ \:\bf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } = 1 \: }}\\ \\ [/tex]
Step-by-step explanation:
Given expression is
[tex]\sf \: = \: {\bigg( x \bigg) }^{ \frac{b}{(b - c)(b - a)} } \times {\bigg( x\bigg) }^{ \frac{c}{(c - a)(c - b)} } \times {\bigg( x \bigg) }^{ \frac{a}{(a - b)(a - c)} } \\ \\ [/tex]
[tex]\qquad\qquad\boxed{ \sf{ \: \because \: {( {x}^{m} )}^{n} = {x}^{mn} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{ \frac{b}{(b - c)(b - a)} + \frac{c}{(c - a)(c - b)} + \frac{a}{(a - b)(a - c)} }\\ \\ [/tex]
[tex]\qquad\qquad\boxed{ \sf{ \: \because \: {x}^{m} \times {x}^{n} = {x}^{m + n} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{ - \frac{b}{(b - c)(a - b)} - \frac{c}{(c - a)(b - c)} - \frac{a}{(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ - b(c - a) - c(a - b) - a(b - c)}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ - bc +ab - ca + bc - ab + ac}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ 0}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{0 }\\ \\ [/tex]
[tex]\sf \: = \: 1\\ \\ [/tex]
Hence,
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ {x}^{0} = 1}\\ \\ \bigstar \: \bf{ {x}^{m} \times {x}^{n} = {x}^{m + n} }\\ \\ \bigstar \: \bf{ {( {x}^{m})}^{n} = {x}^{mn} }\\ \\\bigstar \: \bf{ {x}^{m} \div {x}^{n} = {x}^{m - n} }\\ \\ \bigstar \: \bf{ {x}^{ - n} = \dfrac{1}{ {x}^{n} } }\\ \\\bigstar \: \bf{ {\bigg(\dfrac{a}{b} \bigg) }^{ - n} = {\bigg(\dfrac{b}{a} \bigg) }^{n} }\\ \\\bigstar \: \bf{ {x}^{m} = {x}^{n}\rm\implies \:m = n }\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
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Verified answer
Question :- Simplify the following :
[tex]\sf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } \\ \\ [/tex]
Answer:
[tex]\qquad\boxed{ \sf{ \:\bf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } = 1 \: }}\\ \\ [/tex]
Step-by-step explanation:
Given expression is
[tex]\sf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } \\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{ \frac{b}{(b - c)(b - a)} } \times {\bigg( x\bigg) }^{ \frac{c}{(c - a)(c - b)} } \times {\bigg( x \bigg) }^{ \frac{a}{(a - b)(a - c)} } \\ \\ [/tex]
[tex]\qquad\qquad\boxed{ \sf{ \: \because \: {( {x}^{m} )}^{n} = {x}^{mn} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{ \frac{b}{(b - c)(b - a)} + \frac{c}{(c - a)(c - b)} + \frac{a}{(a - b)(a - c)} }\\ \\ [/tex]
[tex]\qquad\qquad\boxed{ \sf{ \: \because \: {x}^{m} \times {x}^{n} = {x}^{m + n} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{ - \frac{b}{(b - c)(a - b)} - \frac{c}{(c - a)(b - c)} - \frac{a}{(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ - b(c - a) - c(a - b) - a(b - c)}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ - bc +ab - ca + bc - ab + ac}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{\frac{ 0}{(b - c)(a - b)(c - a)} }\\ \\ [/tex]
[tex]\sf \: = \: {\bigg( x \bigg) }^{0 }\\ \\ [/tex]
[tex]\sf \: = \: 1\\ \\ [/tex]
Hence,
[tex]\qquad\boxed{ \sf{ \:\bf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} } = 1 \: }}\\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ {x}^{0} = 1}\\ \\ \bigstar \: \bf{ {x}^{m} \times {x}^{n} = {x}^{m + n} }\\ \\ \bigstar \: \bf{ {( {x}^{m})}^{n} = {x}^{mn} }\\ \\\bigstar \: \bf{ {x}^{m} \div {x}^{n} = {x}^{m - n} }\\ \\ \bigstar \: \bf{ {x}^{ - n} = \dfrac{1}{ {x}^{n} } }\\ \\\bigstar \: \bf{ {\bigg(\dfrac{a}{b} \bigg) }^{ - n} = {\bigg(\dfrac{b}{a} \bigg) }^{n} }\\ \\\bigstar \: \bf{ {x}^{m} = {x}^{n}\rm\implies \:m = n }\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]