Question

Solve ${\left(\frac{x^b}{x^c}\right)}^{b+c-a}{\left(\frac{x^c}{x^a}\right)}^{c+a-b}{\left(\frac{x^a}{x^b}\right)}^{a+b-c}$

Answer 1

${\left(\frac{x^b}{x^c}\right)}^{b+c-a}{\left(\frac{x^c}{x^a}\right)}^{c+a-b}{\left(\frac{x^a}{x^b}\right)}^{a+b-c}$

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

$=\left(x{)}^{(b-c)(b+c)-a(b-c)}\right(x{)}^{(c-a)(c+a)-b(c-a)}(x{)}^{(a-b)(a+b)-c(a-b)}$

$=\left(x{)}^{(b^2-c^2-ab+ac)}\right(x{)}^{(c^2-a^2-bc+ab)}(x{)}^{(a^2-b^2-ca+cb)}$

$=(x{)}^{(b^2-c^2-ab+ac+c^2-a^2-bc+ab+a^2-b^2-ac+bc)}$

$=(x{)}^o$

$=1$