Question

#Sybermath
If $\sqrt[3]{x}+\sqrt[3]{y}=\sqrt[3]{z}$, evaluate $\dfrac{xyz}{(z-x-y)^{3}}$.

Answer 1

$\large\text{ \boxed{\bold{[Given\ information]}} }$

$\large\bullet\ \sqrt[3]{x}+\sqrt[3]{y}=\sqrt[3]{z}$

$\large\text{ \boxed{\bold{[Step\ 1.]}} }$

If we move $\large\sqrt[3]{z}$ and isolate the variables, what will happen?

$\large\cdots\longrightarrow\sqrt[3]{x}+\sqrt[3]{y}-\sqrt[3]{z}=0$

More like, -

$\large\cdots\longrightarrow\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{-z}=0.$

$\large\text{ \boxed{\bold{[Explanation:\ Identity]}} }$

Then, we have learned a very familiar identity of three variables.

$\large\text{ \cdots\longrightarrow\boxed{\begin{aligned}&A^{3}+B^{3}+C^{3}-3ABC\\\\&=(A+B+C)(A^{2}+B^{2}+C^{2}-AB-BC-CA)\end{aligned}} }$

The condition for zero is $\largeA+B+C=0\text{ or }A=B=C$. Also, the given condition exactly matches.

$\large\cdots\longrightarrow A^{3}+B^{3}+C^{3}-3ABC=0$

Then, after simple modification we get $\largeA^{3}+B^{3}+C^{3}=3ABC$.

$\large\text{ \boxed{\bold{[Step\ 2.]}} }$

The condition leads to one equation.

$\large\text{ \cdots\longrightarrow\boxed{x+y-z=-3\sqrt[3]{xyz}} }$

So, by applying the equation, we can finally find out our answer.

$\large\text{ \cdots\longrightarrow\boxed{\begin{aligned}&\dfrac{xyz}{(z-x-y)^{3}}\\\\&=\dfrac{xyz}{-(x+y-z)^{3}}\\\\&=\dfrac{xyz}{-(-3\sqrt[3]{xyz})^{3}}\\\\&=\dfrac{xyz}{27xyz}\\\\&=\underline{\dfrac{1}{27}}\end{aligned}} }$

$\large\text{ \boxed{\bold{[Final\ answer]}} }$

So, we got our answer.

$\large\text{ \cdots\longrightarrow\boxed{\bold{\dfrac{xyz}{(z-x-y)^{3}}=\dfrac{1}{27}}} }$