Please show step-by-step solution

(3)

Given information: $4x^3=2x^{-1}$

To simplify the equation use the operations division and multiplication.

The equation can be written as, $4x^3=\frac{2}{x}$

First of all Multiply by x on both the sides,

$\begin{array}{rcl}\left(4x^3\right)x& =& \left(\frac{2}{x}\right)x\\ 4x^4& =& 2\end{array}$

Divide by 4 on both the sides,

$\begin{array}{rcl}\frac{4x^4}{4}& =& \frac{2}{4}\\ x^4& =& \frac{1}{2}\end{array}$

It is a degree of 4, to simplify this substitute $t=x^2$ and $t^2=x^4$

$t=\sqrt{\frac{1}{2}}, t=-\sqrt{\frac{1}{2}}$

and substitute back $t=x^2$

$x^2=\sqrt{\frac{1}{2}}, x^2=-\sqrt{\frac{1}{2}}$

On solving both, the solution is,

$x=\sqrt[4]{\frac{1}{2}}, x=-\sqrt[4]{\frac{1}{2}} and x=i\left(\sqrt[4]{\frac{1}{2}}\right), x=-i\left(\sqrt[4]{\frac{1}{2}}\right)$