Finding long run total cost function

I am trying to find the long run total cost function, given the firm’s production function $y=L^α K^β$ where $α,β>0$ and two inputs $L$ and $K$ where $ L,K∈R_+^2$, with factor prices $w$ and $r$ where $w,r∈R_{++}^2$

So far, I have tried to solve $min$ $⁡wL+rK s.t.y=L^α K^β$ which then I got $mathcal{L}$$(L,K,λ)=wL+rK+λ(y-L^α K^β)$

Solving for FONC:

$frac{∂L}{∂L}=0⟹w=αλL^{α-1} K^β$

$frac{∂L}{∂K}=0⟹r=βλL^α K^{β-1}$

$frac{∂L}{∂λ}=0⟹ y=L^α K^β$

Then dividing equation 1 by equation 2 I get:

$frac{w}{r}= frac{αλL^{α-1} K^β}{βλL^α K^{β-1} }⟹frac{w}{r}= frac{αK}{βL}⟹L=frac{αr}{βw} K$

However, this is where I run into an issue…

When I substitute $L=frac{αr}{βw} K$ back into equation 3 ($y=L^α K^β$), I get:

$y=(frac{αr}{βw} K)^α K^β$

Which I don’t know how to solve for $K$

I am wondering if I did something wrong, or how do I proceed to the next step, where I can solve for $K$ and $L$ and then proceed to use those values to find the long run total cost function.

Thanks

Pls. do like this
$y=(frac{αr}{βw} K)^α K^β = (frac{αr}{βw})^α K^{(β+α)}$ and then

$$ y (frac{βw}{αr})^α = K^{(β+α)}$$ and finally

$$ y^frac{1}{{(β+α)}} (frac{βw}{αr})^frac{α}{(β+α)} = K := K^star(w,r,y)$$