Cfrgtkky

I have been trying to solve this type of equation for $n$ in terms of $y$: $an^{frac{1}{2}}-bn^{frac{1}{3}}+cn^{frac{1}{6}}=y$ but I have yet to find a way that works. I have also tried several online equation solvers with no avail.

Here is what I have tried so far in solving the following equation for $n$ in terms of $y$:

$2n^{frac{1}{2}}-1.5n^{frac{1}{3}}+n^{frac{1}{6}}=y$

I first tried to rewrite it as a quadratic but after pulling out $n^{frac{1}{6}}$ like this:

$n^{frac{1}{6}}(2n^{frac{1}{3}}-1.5n^{frac{1}{6}}+1)=y$

But then there is nothing further I can do to factor this.

I couldn't find any power that I could raise $n$ to multiply both sides either.

Any advice as how to proceed in solving this would be greatly appreciated.

Thank you.

Let $x=n^{1/6}$ (i.e. $n=x^6$):

$$2x^3-frac 3 2x^2+x=y$$

Subtract $y$ and divide both sides by $2$:

$$x^3-frac 3 4x^2+frac 1 2x-frac 1 2y=0$$

In order to reduce this to a depressed cubic, let $x=z+frac 1 4$:

$$z^3+frac{5z}{16}+frac{3}{32}-frac 1 2y=0$$

In order to turn this into a quadratic, let $z=w+frac{5}{48w}$:

$$w^3+frac{3}{32}-frac 1 2y-frac{125}{110592w^3}=0$$

Multiply by $w^3$:

$$w^6+left(frac{3}{32}-frac 1 2yright)w^3-frac{125}{110592}=0$$

Time to apply the quadratic formula:

$$w^3=frac{frac 1 2 y-frac{3}{32}pmsqrt{left(frac{3}{32}-frac 1 2 yright)^2-frac{125}{27649}}}{frac{125}{55296}}$$

Now, there are three complex solutions to this equation. To represent this, I will use $omega$ to represent a cube root of unity. Thus, the following equation represents three solutions for $w$: One for $omega=1$, one for $omega=frac{-1+sqrt{-3}}{2}$, and one for $omega=frac{-1-sqrt{-3}}{2}$:

$$w=omegasqrt[3]{frac{frac 1 2 y-frac{3}{32}pmsqrt{left(frac{3}{32}-frac 1 2 yright)^2-frac{125}{27649}}}{frac{125}{55296}}}$$

From here, I leave the rest to you: Solve for $z$ using $z=w-frac{5}{48w}$, solve for $x$ using $x=z+frac 1 4$, and, finally, solve for $n$ using $n=x^6$. Good luck finishing the problem!

Using $x=n^6$ as Noble Mushtak did and follow the steps given here, using the hyperbolic solution for one real root, you should arrive to the simple equation
$$color{blue}{x=frac{1}{4}-frac{1}{2} sqrt{frac{5}{3}} sinh left(frac{1}{3} sinh
^{-1}left(frac{1}{192} sqrt{frac{5}{3}} left(frac{81}{4}-108
yright)right)right)}$$ which is positive if $y >0$.