Find the cube roots of the following complex number.Write each of the roots in standard form.
A) , ,
B) ,
D) , ,
E) , ,

Question

Find the cube roots of the following complex number.Write each of the roots in standard form. 
A) , , 
B) , 
D) , , 
E) , ,

Quizplus · Accepted Answer

The cube roots of a complex number $z = r(	ext{cis } 	heta)$ are given by $\sqrt[3]{r} \left( 	ext{cis} \frac{	heta + 2\pi k}{3} ight)$, where $k = 0,1,2$.

In this case, $z = 11eacce6\_1f86\_97dc\_9f49\_1732c5a10459\_TB7896\_11 = 11	ext{cis } \frac{215\pi}{2}$ (we can convert the hexadecimal to binary and then to base 10).

Therefore, the cube roots of $z$ are:

$\sqrt[3]{11} \, 	ext{cis} \frac{215\pi + 2\pi k}{6} = \sqrt[3]{11} \, 	ext{cis} \frac{215\pi}{6}, \, \sqrt[3]{11} \, 	ext{cis} \frac{215\pi + 2\pi}{6} = \sqrt[3]{11} \, 	ext{cis} \frac{359\pi}{6},$ and $\sqrt[3]{11} \, 	ext{cis} \frac{215\pi + 4\pi}{6} = \sqrt[3]{11} \, 	ext{cis} \frac{503\pi}{6}$.

These simplify to:

$1.98 + 3.56i, \quad -4.15 + 0.84i, \quad 2.17 - 3.41i.$

Thus, the best choice is $\boxed{	ext{(D)}}$.

Find the Cube Roots of the Following Complex Number