Question

$( \frac{5 ^{a} }{5 ^{b} }) ^{a + b} ( \frac{5 ^{b} }{5 ^{c} }) ^{b + c} ( \frac{5 ^{c} }{5 ^{a} }) ^{c + a} =$
please answer with explanation​

Answer 1

$( \frac{5 ^{a} }{5 ^{b} }) ^{a + b} ( \frac{5 ^{b} }{5 ^{c} }) ^{b + c} ( \frac{5 ^{c} }{5 ^{a} }) ^{c + a}$

$= ( {5}^{a - b} ) {}^{a + b} ( {5}^{ b - c} ) {}^{c + b} ( {5}^{c - a} ) {}^{a + c}$

$= ( {5}^{ {a}^{2} - {b}^{2} } )( {5}^{ {b}^{2} - {c}^{2} } )( {5}^{ {c}^{2} - {a}^{2} } )$

$= {5}^{ {a}^{2} - {b}^{2} + {b}^{2} - {c}^{2} + {c}^{2} - {a}^{2} }$

$= {5}^{ \cancel{{a}^{2} } - \cancel{{b}^{2} }+ \cancel{{b}^{2} } - \cancel{ {c}^{2}} +\cancel{ {c}^{2} } - \cancel{{a}^{2}} }$

$= {5}^{0}$

$= 1$