Find the cube roots of the following complex number. Write each of the roots in standard form.

Question

Quizplus · Accepted Answer

To find the cube roots of a complex number, we need to first express it in polar form. Let's write the given complex number in polar form:

|r| = sqrt(11eb80b1^2 + af8c^2 + 829c^2 + a291^2 + 03a51331bf73^2 + TB8633^2 + 11^2) = sqrt(3)

arg(r) = arctan(imaginary part/real part) = arctan(-11eb80b1/af8c) = -1.9635 radians

So, the polar form of the complex number is:

r = sqrt(3) cis (-1.9635)

Now, we can find the cube roots by using the formula:

sqrt(3)^(1/3) cis (-1.9635/3 + 2πk/3)

where k = 0, 1, 2 (for the three cube roots)

Simplifying, we get:

r^(1/3) = (sqrt(3)/2)^(1/3) cis (-0.6545 + 2πk/3)

Now, we substitute k = 0, 1, 2 to get the three cube roots:

r^(1/3) = (sqrt(3)/2)^(1/3) cis (-0.6545)
r^(1/3) = (sqrt(3)/2)^(1/3) cis (1.3089)
r^(1/3) = (sqrt(3)/2)^(1/3) cis (-2.9634)

Converting to standard form, we get:

r^(1/3) = 11eb80b1_af8c_c8b2_a291_a6832c5b5bf5_TB8633_11
r^(1/3) = 11eb80b1_af8c_351c_a291_0cd424e6143c_TB8633_11
r^(1/3) = 11eb80b1_af8c_7f38_a291_0c7eaf15dbd1_TB8633_11

Only choice (D) matches these values.

Find the Cube Roots of the Following Complex Number