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[tex] \sf \bf \orange{Prove \: \bf \red{That:}}[/tex]

[tex]\sf\left(\dfrac{x^{a^2+b^2}}{x^{ab}}\right)^{a+b}\left(\dfrac{x^{b^2+c^2}}{x^{bc}}\right)^{b+c}\left(\dfrac{x^{c^2+a^2}}{x^{ca}}\right)^{c+a}=x^{2\left(a^3+b^3+c^3\right)}[/tex]

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Sagot :

[tex]\sf\left(\dfrac{x^{a^2+b^2}}{x^{ab}}\right)^{a+b}\left(\dfrac{x^{b^2+c^2}}{x^{bc}}\right)^{b+c}\left(\dfrac{x^{c^2+a^2}}{x^{ca}}\right)^{c+a} \\ = {x}^{ ({a}^{2} + {b}^{2} - ab)(a + b)} \: \: {x}^{ ({b}^{2} + {c}^{2} - bc)( b + c)} \: \: {x}^{ ({c}^{2} + {a}^{2} - ca)(c + a)} \\ = {x}^{ {a}^{3} + {b}^{3 } - {a}^{2}b + {a}^{2} b + {b}^{3} - a {b}^{2} } \: \: {x}^{ {b}^{3} + {c}^{3 } - {b}^{2}c + {b}^{2} c + {c}^{3} - b{c}^{2} } \: \: {x}^{ {c}^{3} + {a}^{3 } - {c}^{2}a + {c}^{2} a + {c}^{3} - c {a}^{2} } \: \: \\ = {x}^{{a}^{3} + {b}^{3 } - {a}^{2}b + {a}^{2} b + {b}^{3} - a {b}^{2} + {b}^{3} + {c}^{3 } - {b}^{2}c + {b}^{2} c + {c}^{3} - b{c}^{2} + {c}^{3} + {a}^{3 } - {c}^{2}a + {c}^{2} a + {c}^{3} - c {a}^{2}} \\

= {x}^{ {a}^{3} + {b}^{3} } \: \: {x}^{ {b}^{3} + {c}^{3} } \: \: {x}^{ {c}^{3} + {a}^{3} } \\ = {x}^{{a}^{3} + {b}^{3} + {b}^{3} + {c}^{3} + {c}^{3} + {a}^{3}}

\\ = {x}^{2 {a}^{3} + 2 {b}^{3} + 2 {c}^{3} } \\ = {x}^{2( {a}^{3} + {b}^{3} + {c}^{3} } [/tex]

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