Question:- Evaluate
[tex]\sf \: \sqrt[a + b]{\dfrac{ {x}^{ {a}^{2} } }{ {x}^{ {b}^{2} } } } \times \sqrt[b + c]{\dfrac{ {x}^{ {b}^{2} } }{ {x}^{ {c}^{2} } } } \times \sqrt[c + a]{\dfrac{ {x}^{ {c}^{2} } }{ {x}^{ {a}^{2} } } } \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
We know,
[tex]\boxed{ \sf{ \:\dfrac{ {x}^{m} }{ {x}^{n} } = {x}^{m - n} \: }} \\ \\ [/tex]
So, using this result, we get
[tex]\sf \: = \: \sqrt[a + b]{ {x}^{ {a}^{2} - {b}^{2} } } \times \sqrt[b + c]{ {x}^{ {b}^{2} - {c}^{2} } } \times \sqrt[c + a]{ {x}^{ {c}^{2} - {a}^{2} } } \\ \\ [/tex]
[tex]\boxed{ \sf{ \: \sqrt[n]{x} = {\bigg(x \bigg) }^{\dfrac{1}{n} } \: }} \\ \\ [/tex]
[tex]\sf \: = \: {\bigg(x \bigg) }^{\dfrac{ {a}^{2} - {b}^{2} }{a + b} } \times {\bigg(x \bigg) }^{\dfrac{ {b}^{2} - {c}^{2} }{b - c} } \times {\bigg(x \bigg) }^{\dfrac{ {c}^{2} - {a}^{2} }{c - a} } \\ \\ [/tex]
[tex]\sf \: = \: {\bigg(x \bigg) }^{\dfrac{(a + b)(a - b)}{a + b} } \times {\bigg(x \bigg) }^{\dfrac{(b - c)(b + c) }{b - c} } \times {\bigg(x \bigg) }^{\dfrac{(c - a)(c + a)}{c - a} } \\ \\ [/tex]
[tex]\sf \: = \: {x}^{a - b} \times {x}^{b - c} \times {x}^{c - a} \\ \\ [/tex]
[tex]\sf \: = \: {x}^{a - b + b - c + c - a} \\ \\ [/tex]
[tex]\sf \: = \: {x}^{0} \\ \\ [/tex]
[tex]\sf \: = \: 1 \\ \\ [/tex]
Hence,
[tex]\sf \: \bf\implies \: \sqrt[a + b]{\dfrac{ {x}^{ {a}^{2} } }{ {x}^{ {b}^{2} } } } \times \sqrt[b + c]{\dfrac{ {x}^{ {b}^{2} } }{ {x}^{ {c}^{2} } } } \times \sqrt[c + a]{\dfrac{ {x}^{ {c}^{2} } }{ {x}^{ {a}^{2} } } } = 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]
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Verified answer
Question:- Evaluate
[tex]\sf \: \sqrt[a + b]{\dfrac{ {x}^{ {a}^{2} } }{ {x}^{ {b}^{2} } } } \times \sqrt[b + c]{\dfrac{ {x}^{ {b}^{2} } }{ {x}^{ {c}^{2} } } } \times \sqrt[c + a]{\dfrac{ {x}^{ {c}^{2} } }{ {x}^{ {a}^{2} } } } \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: \sqrt[a + b]{\dfrac{ {x}^{ {a}^{2} } }{ {x}^{ {b}^{2} } } } \times \sqrt[b + c]{\dfrac{ {x}^{ {b}^{2} } }{ {x}^{ {c}^{2} } } } \times \sqrt[c + a]{\dfrac{ {x}^{ {c}^{2} } }{ {x}^{ {a}^{2} } } } \\ \\ \\ [/tex]
We know,
[tex]\boxed{ \sf{ \:\dfrac{ {x}^{m} }{ {x}^{n} } = {x}^{m - n} \: }} \\ \\ [/tex]
So, using this result, we get
[tex]\sf \: = \: \sqrt[a + b]{ {x}^{ {a}^{2} - {b}^{2} } } \times \sqrt[b + c]{ {x}^{ {b}^{2} - {c}^{2} } } \times \sqrt[c + a]{ {x}^{ {c}^{2} - {a}^{2} } } \\ \\ [/tex]
We know,
[tex]\boxed{ \sf{ \: \sqrt[n]{x} = {\bigg(x \bigg) }^{\dfrac{1}{n} } \: }} \\ \\ [/tex]
So, using this result, we get
[tex]\sf \: = \: {\bigg(x \bigg) }^{\dfrac{ {a}^{2} - {b}^{2} }{a + b} } \times {\bigg(x \bigg) }^{\dfrac{ {b}^{2} - {c}^{2} }{b - c} } \times {\bigg(x \bigg) }^{\dfrac{ {c}^{2} - {a}^{2} }{c - a} } \\ \\ [/tex]
[tex]\sf \: = \: {\bigg(x \bigg) }^{\dfrac{(a + b)(a - b)}{a + b} } \times {\bigg(x \bigg) }^{\dfrac{(b - c)(b + c) }{b - c} } \times {\bigg(x \bigg) }^{\dfrac{(c - a)(c + a)}{c - a} } \\ \\ [/tex]
[tex]\sf \: = \: {x}^{a - b} \times {x}^{b - c} \times {x}^{c - a} \\ \\ [/tex]
[tex]\sf \: = \: {x}^{a - b + b - c + c - a} \\ \\ [/tex]
[tex]\sf \: = \: {x}^{0} \\ \\ [/tex]
[tex]\sf \: = \: 1 \\ \\ [/tex]
Hence,
[tex]\sf \: \bf\implies \: \sqrt[a + b]{\dfrac{ {x}^{ {a}^{2} } }{ {x}^{ {b}^{2} } } } \times \sqrt[b + c]{\dfrac{ {x}^{ {b}^{2} } }{ {x}^{ {c}^{2} } } } \times \sqrt[c + a]{\dfrac{ {x}^{ {c}^{2} } }{ {x}^{ {a}^{2} } } } = 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]