Find the indicated roots and graph them in the complex plane. The fifth roots of 32 Im W1 i Wo W2 -2 -1 W2 -2 W3 W1 W3 Im WA Re 2 Wo i -2 W2 Im W1 i W3 Im W1 W2 W2 Wo Re 2 W4 Wo Re Re -2 2 2 -1 W4 W3 W4 Referring to the labels from the graph you selected above, give the exact roots. (Express
in radians.)
The Correct Answer and Explanation is:
To find the fifth roots of 323232, express the number in polar (trigonometric) form. The number 32 is a real number with no imaginary part:32=32(cos(0)+isin(0))32 = 32(\cos(0) + i\sin(0))32=32(cos(0)+isin(0))
Let the complex number be z=r(cos(θ)+isin(θ))z = r(\cos(\theta) + i\sin(\theta))z=r(cos(θ)+isin(θ)), where r=32r = 32r=32 and θ=0\theta = 0θ=0.
To find the fifth roots of a complex number, apply De Moivre’s Theorem:wk=325(cos(2πk5)+isin(2πk5)),k=0,1,2,3,4w_k = \sqrt[5]{32} \left( \cos\left(\frac{2\pi k}{5}\right) + i\sin\left(\frac{2\pi k}{5}\right) \right), \quad k = 0, 1, 2, 3, 4wk=532(cos(52πk)+isin(52πk)),k=0,1,2,3,4
Since 325=2\sqrt[5]{32} = 2532=2, the roots are:
- w0=2(cos(2π(0)5)+isin(2π(0)5))=2(cos(0)+isin(0))=2w_0 = 2\left( \cos\left(\frac{2\pi(0)}{5} \right) + i\sin\left( \frac{2\pi(0)}{5} \right) \right) = 2(\cos(0) + i\sin(0)) = 2w0=2(cos(52π(0))+isin(52π(0)))=2(cos(0)+isin(0))=2
- w1=2(cos(2π5)+isin(2π5))w_1 = 2\left( \cos\left(\frac{2\pi}{5} \right) + i\sin\left( \frac{2\pi}{5} \right) \right)w1=2(cos(52π)+isin(52π))
- w2=2(cos(4π5)+isin(4π5))w_2 = 2\left( \cos\left(\frac{4\pi}{5} \right) + i\sin\left( \frac{4\pi}{5} \right) \right)w2=2(cos(54π)+isin(54π))
- w3=2(cos(6π5)+isin(6π5))w_3 = 2\left( \cos\left(\frac{6\pi}{5} \right) + i\sin\left( \frac{6\pi}{5} \right) \right)w3=2(cos(56π)+isin(56π))
- w4=2(cos(8π5)+isin(8π5))w_4 = 2\left( \cos\left(\frac{8\pi}{5} \right) + i\sin\left( \frac{8\pi}{5} \right) \right)w4=2(cos(58π)+isin(58π))
From the graph selected (bottom right), the roots appear labeled correctly in order around the unit circle:
Exact roots:
- w0=2(cos(0π5)+isin(0π5))w_0 = 2\left( \cos\left(\frac{0\pi}{5}\right) + i\sin\left(\frac{0\pi}{5}\right) \right)w0=2(cos(50π)+isin(50π))
- w1=2(cos(2π5)+isin(2π5))w_1 = 2\left( \cos\left(\frac{2\pi}{5}\right) + i\sin\left(\frac{2\pi}{5}\right) \right)w1=2(cos(52π)+isin(52π))
- w2=2(cos(4π5)+isin(4π5))w_2 = 2\left( \cos\left(\frac{4\pi}{5}\right) + i\sin\left(\frac{4\pi}{5}\right) \right)w2=2(cos(54π)+isin(54π))
- w3=2(cos(6π5)+isin(6π5))w_3 = 2\left( \cos\left(\frac{6\pi}{5}\right) + i\sin\left(\frac{6\pi}{5}\right) \right)w3=2(cos(56π)+isin(56π))
- w4=2(cos(8π5)+isin(8π5))w_4 = 2\left( \cos\left(\frac{8\pi}{5}\right) + i\sin\left(\frac{8\pi}{5}\right) \right)w4=2(cos(58π)+isin(58π))
These roots are evenly spaced around the origin in the complex plane, forming a regular pentagon. Each root lies at an angle increment of 2π5\frac{2\pi}{5}52π radians from the previous one, creating symmetry.
