Question

Let $a+b+c=0$, then find the value of $\sqrt[bc]{bc\frac{x^{a^2}}{x^{bc}}}$$\times \sqrt[ca]{ca\frac{x^{b^2}}{x^{ca}}}$$\times \sqrt[ab]{ab\frac{x^{c^2}}{x^{ab}}}$.

Answer 1

Given,

$a+b+c=0$

or, $a+b=-c$.....(1).

Cubing both sides we get,

$a+b{)}^3=-c^3$

or, $a^3+b^3+3ab(a+b)=-c^3$

or, $a^3+b^3+c^3=3abc$.......(2) [ Using (1)].

Now,

$\sqrt[bc]{bc\frac{x^{a^2}}{x^{bc}}}$$\times \sqrt[ca]{ca\frac{x^{b^2}}{x^{ca}}}$$\times \sqrt[ab]{ab\frac{x^{c^2}}{x^{ab}}}$

$=\sqrt[bc]{bc{x}^{a^2-bc}}$$\times \sqrt[ca]{ca{x}^{b^2-ac}}$$\times \sqrt[ab]{ab{x}^{c^2-ab}}$

$=x^{(\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}-3)}$

$=x^{\frac{a^3+b^3+c^3-3abc}{abc}}$

$x^{3abc-3abc}$

$=x^0$ [ Using (2)]

$=1$.