# How do you solve: x^3/2=125?

$x = 5 \sqrt[3]{2} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 \setminus \pi}{3}} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{4 \setminus \pi}{3}}$

#### Explanation:

${x}^{3} / 2 = 125$

${x}^{3} = 125 \setminus \cdot 2$

${x}^{3} = 250$

${x}^{3} = 250 {e}^{i 0}$

${x}^{3} = 250 {e}^{i 2 k \setminus \pi}$

$x = {\left(250 {e}^{i 2 k \setminus \pi}\right)}^{\frac{1}{3}}$

$x = \setminus \sqrt[3]{250} {e}^{i \frac{2 k \setminus \pi}{3}}$

$x = 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 k \setminus \pi}{3}}$

Where, $k = 0 , 1 , 2$

Now, setting the values of $k$, we get three roots of given cubic equation as follows

$x = 5 \sqrt[3]{2} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 \setminus \pi}{3}} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{4 \setminus \pi}{3}}$