How do I convert between these two equivalent expressions?

Question

While working on some recreational mathematics involving the cubic formula, I noticed that the expression $\frac{8}{\sqrt[3]{19 - \sqrt{297}}}$ is equivalent to the expression $2 × \sqrt[3]{19 + \sqrt{297}}$. Knowing they are equivalent, I feel like there should be a way to algebraically manipulate either expression to get the other. However, I can't figure it out. Can someone explain how to do this?

Edit: Further experimentation with the cubic formula revealed that the relation in question can be generalized as $\frac{a}{\sqrt[3]{b - \sqrt{b^2 - a^3}}} = \sqrt[3]{b + \sqrt{b^2 - a^3}}$, with values of $a = 4$ and $b = 19$ giving the expressions with which I started.

Answer 1

From the comments, I see that I can rewrite $\frac{a}{\sqrt[3]{b - \sqrt{b^2 - a^3}}} = \sqrt[3]{b + \sqrt{b^2 - a^3}}$ by moving the $a$ inside the cube root to get $\sqrt[3]{\frac{a^3}{b - \sqrt{b^2 - a^3}}} = \sqrt[3]{b + \sqrt{b^2 - a^3}}$.

Get rid of the cube roots by cubing each side to get $\frac{a^3}{b - \sqrt{b^2 - a^3}} = b + \sqrt{b^2 - a^3}$.

Multiply both sides by $b - \sqrt{b^2 - a^3}$ to get $a^3 = (b + \sqrt{b^2 - a^3}) × (b - \sqrt{b^2 - a^3})$.

Use the identity $(x + y) × (x - y) = x^2 - y^2$ on the right side, eliminating the square root to get $a^3 = b^2 - (b^2 - a^3)$.

Expand the parenthetical and cancel the $b$s to get $a^3 = a^3$, proving the equality.

Answer 2

It is common to eliminate the square root by using the identity $(a-b)(a+b) = a^2 - b^2$, so multipling both nominator and denominator by $\sqrt[3]{(19 + \sqrt{297})}$ will get

$$ \dfrac{(8\times\sqrt[3]{(19 + \sqrt{297})})}{\sqrt[3]{19^2-(\sqrt{297})^2}} = \dfrac{(8\times\sqrt[3]{(19 + \sqrt{297})})}{\sqrt[3]{64}} = 2\times\sqrt[3]{(19 + \sqrt{297})} $$

Answer 3

$19^2-279=64.$ So $(19-\sqrt {279}\,)(19+\sqrt {279}\,)=64.$

So $19+\sqrt {279}\,=\frac {64}{19-\sqrt {279}}.$

So $2\cdot \sqrt[3] {19+\sqrt {279}} =2\cdot \sqrt [3] \frac {64}{19-\sqrt {279}}=$

$=2\cdot 4\cdot \frac {1}{ \sqrt [3]{19-\sqrt {279}}}.$