Resolve The fraction
[tex] \\ \tt\[ \frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{a}{c}+\frac{c}{a}+3 \][/tex]
[tex] \rule{300pt}{0.1pt}[/tex]
[tex] \texttt{Answer: \( \tt (a b+b c+a c)(a+b+c) / a b c \)}[/tex]
Step-by-step explanation:
[tex]\[ \begin{array}{l} \\ \\ \\ \displaystyle\bf \qquad\[ \frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{a}{c}+\frac{c}{a}+3 \] \\ \\ \\ \\ \displaystyle\bf \qquad=\Rightarrow \frac{\left(a^{2}+b^{2}\right) }{ a b}+ \frac{\left(b^{2}+c^{2}\right)}{ b c}+ \frac{\left(c^{2}+a^{2}\right) }{a c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(c\left(a^{2}+b^{2}\right)+a\left(b^{2}+c^{2}\right)+b\left(a^{2}+c^{2}\right)\right)}{ a b c}+3 \\ \\ \\ \\ \displaystyle\bf \qquad=\Rightarrow \frac{\left(a^{2} c+b^{2} c+a b^{2}+a c^{2}+a^{2} b+c^{2} b\right) }{ a b c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+b^{2} c+b c^{2}+a^{2} b+a b^{2}\right) }{a b c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+b^{2} c+b c^{2}+a^{2} b+a b^{2}+3 a b c\right) }{a b c} \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+a b c+b^{2} c+b c^{2}+a b c+a^{2} b+a b^{2}+a b c\right) }{a b c} \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{(a c(a+c+b)+b c(b+c+a)+a b(a+b+c)) }{a b c }\\\\ \\ \\ \fcolorbox{red}{pink}{ \boxed{ \color{blue} \displaystyle\bf \qquad =\Rightarrow \frac{(a b+b c+a c)(a+b+c) }{a b c}}} \end{array} \] [/tex]
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Question :-
Resolve The fraction
[tex] \\ \tt\[ \frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{a}{c}+\frac{c}{a}+3 \][/tex]
[tex] \rule{300pt}{0.1pt}[/tex]
[tex] \texttt{Answer: \( \tt (a b+b c+a c)(a+b+c) / a b c \)}[/tex]
[tex] \rule{300pt}{0.1pt}[/tex]
Step-by-step explanation:
[tex]\[ \begin{array}{l} \\ \\ \\ \displaystyle\bf \qquad\[ \frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{a}{c}+\frac{c}{a}+3 \] \\ \\ \\ \\ \displaystyle\bf \qquad=\Rightarrow \frac{\left(a^{2}+b^{2}\right) }{ a b}+ \frac{\left(b^{2}+c^{2}\right)}{ b c}+ \frac{\left(c^{2}+a^{2}\right) }{a c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(c\left(a^{2}+b^{2}\right)+a\left(b^{2}+c^{2}\right)+b\left(a^{2}+c^{2}\right)\right)}{ a b c}+3 \\ \\ \\ \\ \displaystyle\bf \qquad=\Rightarrow \frac{\left(a^{2} c+b^{2} c+a b^{2}+a c^{2}+a^{2} b+c^{2} b\right) }{ a b c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+b^{2} c+b c^{2}+a^{2} b+a b^{2}\right) }{a b c}+3 \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+b^{2} c+b c^{2}+a^{2} b+a b^{2}+3 a b c\right) }{a b c} \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{\left(a^{2} c+a c^{2}+a b c+b^{2} c+b c^{2}+a b c+a^{2} b+a b^{2}+a b c\right) }{a b c} \\\\ \\ \\ \displaystyle\bf \qquad =\Rightarrow \frac{(a c(a+c+b)+b c(b+c+a)+a b(a+b+c)) }{a b c }\\\\ \\ \\ \fcolorbox{red}{pink}{ \boxed{ \color{blue} \displaystyle\bf \qquad =\Rightarrow \frac{(a b+b c+a c)(a+b+c) }{a b c}}} \end{array} \] [/tex]
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