Question

simplify 1 divided by 1 + x power b minus a + x power c minus a + 1 / 1 + x power a minus b + x power c minus b + 1 / 1 + x power a minus c plus x power b minus c

Answer 1

SOLUTION

TO SIMPLIFY

$\displaystyle \sf{ \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{a - c} + {x}^{b- c} } \: }$

FORMULA TO BE IMPLEMENTED

We are aware of the identity of indices that

$\displaystyle \sf{ \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n} \: }$

EVALUATION

$\displaystyle \sf{ \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{a - c} + {x}^{b- c} } \: }$

$= \displaystyle \sf{ \frac{1}{1 + \frac{ {x}^{b} }{ {x}^{a} } +\frac{ {x}^{c} }{ {x}^{a} } } + \frac{1}{1 + \frac{ {x}^{a} }{ {x}^{b} } +\frac{ {x}^{c} }{ {x}^{b} } } + \frac{1}{1 + \frac{ {x}^{a} }{ {x}^{c} } +\frac{ {x}^{b} }{ {x}^{c} } }}$

$= \displaystyle \sf{ \frac{1}{ \frac{ {x}^{a} + {x}^{b} + {x}^{c} }{ {x}^{a} } } + \frac{1}{ \frac{ {x}^{b} + {x}^{a} + {x}^{c} }{ {x}^{b} } } + \frac{1}{ \frac{ {x}^{c} + {x}^{a} + {x}^{b} }{ {x}^{c} } } }$

$\displaystyle \sf{ = \frac{ {x}^{a} }{ {x}^{a} + {x}^{b} + {x}^{c} } + \frac{ {x}^{b} }{ {x}^{a} + {x}^{b} + {x}^{c} } + \frac{ {x}^{c} }{ {x}^{a} + {x}^{b} + {x}^{c} } }$

$\displaystyle \sf{ = \frac{ {x}^{a} + {x}^{b} + {x}^{c} }{ {x}^{a} + {x}^{b} + {x}^{c} } }$

$= \sf{1}$

FINAL ANSWER

$\displaystyle \sf{ \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{a - c} + {x}^{b- c} } \: } = 1$

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