A) 1
B) \[{{(a+b+c)}^{3}}\]
C) \[{{\text{a}}^{\text{2}}}\text{+}{{\text{b}}^{\text{2}}}\text{+}{{\text{c}}^{\text{2}}}\]
D) 0
Correct Answer: A
Solution :
\[{{\left( \frac{{{x}^{a}}}{{{x}^{b}}} \right)}^{{{a}^{2}}+{{b}^{2}}+ab}}\times {{\left( \frac{{{x}^{b}}}{{{x}^{c}}} \right)}^{{{b}^{2}}+{{c}^{2}}+cb}}\times {{\left( \frac{{{x}^{c}}}{{{x}^{a}}} \right)}^{{{c}^{2}}+{{a}^{2}}+ca}}\] \[={{({{x}^{a-b}})}^{{{a}^{2}}+{{b}^{2}}+ab}}\times {{({{x}^{b-c}})}^{{{b}^{2}}+{{c}^{2}}+cb}}\times {{({{x}^{c-a}})}^{{{c}^{2}}+{{a}^{2}}+ca}}\] \[={{x}^{(a-b)({{a}^{2}}+{{b}^{2}}+ab)+(b-c)({{b}^{2}}+{{c}^{2}}+cb)+(c-a)({{c}^{2}}+{{a}^{2}}+ca)}}\] \[={{x}^{({{a}^{3}}-{{b}^{3}}+{{b}^{3}}-{{c}^{3}}+{{c}^{3}}-{{a}^{3}})={{x}^{0}}=1}}\].
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