Question

Hello people, can anybody solve it ​

Answer 1

$\\ \rm\Rrightarrow \left(\dfrac{x^a}{x^{-b}}\right)^{a^2-ab+b^2}\times \left(\dfrac{x^b}{x^{-c}}\right)^{b^2-bc+c^2}\times \left(\dfrac{x^c}{x^{-a}}\right)^{c^2-ca+a^2}=1$

$\\ \rm\Rrightarrow (x^{a+b})^{a^2-ab+b^2}\times (x^{b+c})^{b^2-bc+c^2}\times (x^{c+a})^{c^2-ac+a^2}=1$

$\\ \rm\Rrightarrow x^{(a+b)(a^2-ab+b^2)}\times x^{(b+c)(b^2-bc+c^2)}\times x^{(c+a)(c^2-ca+a^2)}=1$

• a³+b³=(a+b)(a²-ab+b²)

$\\ \rm\Rrightarrow x^{a^3+b^3}\times x^{b^3+c^3}\times x^{c^3+a^3}=1$

$\\ \rm\Rrightarrow x^{a^3+b^3+b^3+c^3+c^3+a^3}=1$

$\\ \rm\Rrightarrow x^{2a^3+2b^3+2c^3}=x^0$

$\\ \rm\Rrightarrow 2(a^3+b^3+c^3)=0$

$\\ \rm\Rrightarrow a^3+b^3+c^3=0$