Question

show that (x^a+b)² (x^b+c)² (x^c+a²) / (x^a×x^b×x^c)² = 1


❌( don't give unnecessary answers ) ❌☺️( give correct answer only with full explanation ) ☺️​​

Answer 1

Appropriate Question :-

Prove that :-

$\rm :\longmapsto\:\dfrac{ { {(x}^{a + b} )} \: ^{2} \times { {(x}^{b + c} )} \: ^{2} \times { {(x}^{c + a} )} \: ^{2} }{ {( {x}^{a} \times {x}^{b} \times {x}^{c})}^{4} } = 1$

$\large\underline{\sf{Solution-}}$

Basic Identities Used :-

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}} \\ \end{gathered}$

Let's solve the problem now!!!

Given that,

$\rm :\longmapsto\:\dfrac{ { {(x}^{a + b} )} \: ^{2} \times { {(x}^{b + c} )} \: ^{2} \times { {(x}^{c + a} )} \: ^{2} }{ {( {x}^{a} \times {x}^{b} \times {x}^{c})}^{4} }$

$\rm \: \: = \: \: \:\dfrac{ { {(x}^{2a + 2b} )} \: \times { {(x}^{2b + 2c} )} \: \times { {(x}^{2c + 2a} )} \:}{ {( {x}^{a + b + c})}^{4} }$

$\rm \: \: = \: \: \:\dfrac{ { {x}^{2a + 2b + 2b + 2c + 2c + 2a} } \: \:}{ {( {x}^{4a + 4b + 4c})}}$

$\rm \: \: = \: \: \:\dfrac{ { {x}^{4a + 4b + 4c } } \: \:}{ {{x}^{4a + 4b + 4c}}}$

$\rm \: \: = \: \: 1$