Question

Evaluate the following cube root
$\sqrt[3]{-2{\displaystyle \frac{10}{27}}}$

Answer 1

Hint:- In this question we have given $\sqrt[3]{-2{\displaystyle \frac{10}{27}}}$ So the key concept is that first simplify the inner part of the cube root by using the concept of fraction and then apply the cube root.

Complete step-by-step answer:
Now consider the given expression
$$\begin{gathered} \Rightarrow \sqrt[3]{-2{\displaystyle \frac{10}{27}}} \\ \Rightarrow \sqrt[3]{-{\displaystyle \frac{64}{27}}} \\ \Rightarrow \sqrt[3]{-{\left({\displaystyle \frac{4}{3}}\right)}^3} \\ \Rightarrow \sqrt[3]{(-1)\times {\left({\displaystyle \frac{4}{3}}\right)}^3} \end{gathered}$$ ………… (1)
Now taken ${\displaystyle \frac{4}{3}}$ outside of cube root from equation (1) we get,
$\Rightarrow {\displaystyle \frac{4}{3}}\times \sqrt[3]{-1}$ ………. (2)
And we can write $-1={\left(-1\right)}^3$ in equation (2) we get,
$$\begin{gathered} \Rightarrow {\displaystyle \frac{4}{3}}\times \sqrt[3]{{\left(-1\right)}^3} \\ \Rightarrow {\displaystyle \frac{4}{3}}\times (-1) \\ \Rightarrow -{\displaystyle \frac{4}{3}} \end{gathered}$$
 Hence the answer is $-{\displaystyle \frac{4}{3}}$ .
Note :- Whenever we face such types of problems the key concept is to simplify the given expression in-to-out which means that first solve the inner part and then simplify the cube root part of the expression to get the right answer.