Is there a formula for $\sqrt[3]{a}+\sqrt[3]{b}=\sqrt[3]{?}$ similar to $\sqrt{a}+\sqrt{b}=\sqrt{a+b+2\sqrt{ab}}\;$? I know this ancient formula:
$$\sqrt{a}+\sqrt{b}=\sqrt{a+b+2\sqrt{ab}}$$ And I would like to know if there is a similar formula for:
$$\sqrt[3]{a}+\sqrt[3]{b}=\sqrt[3]{?}$$
I can't find it. Thank you
Notice that your initial formula is nothing more than $(\sqrt{a}+\sqrt{b})^2 = a + b + 2\sqrt{ab}$.
Thus, starting from $(\sqrt[3]{a}+\sqrt[3]b)^3 = a + b + 3\sqrt[3]{ab} \ (\sqrt[3]{a}+\sqrt[3]{b})$, you can write (by taking cubic roots on both sides)
$$\sqrt[3]{a}+\sqrt[3]b = \sqrt[3]{a + b + 3\sqrt[3]{ab} \ (\sqrt[3]{a}+\sqrt[3]{b})}.$$
To @Babado (@Z Ahmed , @Mostafa Ayaz)
I already knew that formula, but what I want to find out is in the answer.I don’t understand how that works.
If I take cubic roots on both sides, then the answer brings me nothing. In the first formula I can know and operate, for example:
$\sqrt{ab}$
, but no:
$\sqrt{a}+\sqrt{b}$