How do I solve an equation with three terms, with the unknown inside a square root, inside a third root, in two of them?

Question

The equation is $$\\ \sqrt[3]{\sqrt{a}+b}+\sqrt[3]{-\sqrt{a}+b}=k.$$

How do I find$\ a$?

Answer 1

$$k=(b+c)^{1/3}+(b-c)^{1/3}$$ where $c=\sqrt a$. We use $(x+y)^3=x^3+y^3+3xy(x+y)$.

$$k^3=b+c+b-c+3(b^2-c^2)^{1/3}k=2b+3k(b^2-c^2)^{1/3}$$

Now subtract $2b$, divide by $3k$, cube, subtract from $b^2$, and you have $c^2$, which is $a$.

Answer 2

There are some nice cancellations. It will look less imposing if we let $c=\sqrt a$ Then we have $$\sqrt[3]{c+b}+\sqrt[3]{-c+b}=k\\ (c+b)+(b-c)+3\sqrt[3]{2b(b^2-c^2)}=k^3\\3\sqrt[3]{2b(b^2-c^2)}=k^3-2b$$ an you are on your way.