Question

Solve the following equations:
$3x^{\frac{1}{2n}}-x^{\frac{1}{n}}-2=0$.

Answer 1

$3x^{\frac{1}{2n}}-x^{\frac{1}{n}}-2=03x^{\frac{1}{2n}}-{\left(x^{\frac{1}{2n}}\right)}^2-2=0$

Put $x^{\frac{1}{2n}}=t$
$3t-t^2-2=0t^2-3t+2=0t^2-t-2t+2=0t(t-2)-2(t-1)=0(t-1)(t-2)=0t=1,2x^{\frac{1}{2n}}=t\Rightarrow x=t^{2n}x=1^{2n},2^{2n}x=1,2^{2n}$