Find the fifth roots of 243 complex numbers

Find the fifth roots of 243 complex numbers.

The correct answer and explanation is:

The problem asks for the fifth roots of 243, a real number. To find the fifth roots of 243 in the complex plane, follow these steps:

Step 1: Express 243 in polar form
243 is a positive real number. Its polar form is:
r=243r = 243, θ=0∘\theta = 0^\circ (or 0 radians) since it lies on the positive real axis.

Step 2: Find the magnitude of the fifth root
The magnitude rr of the root is given by the fifth root of 243: ∣z∣=2435|z| = \sqrt[5]{243}

Since 243=35243 = 3^5, ∣z∣=355=3|z| = \sqrt[5]{3^5} = 3

Step 3: Find the arguments of the fifth roots
The fifth roots of a complex number have arguments (angles) spaced evenly around the unit circle, separated by 2π5\frac{2\pi}{5} radians (or 72 degrees).

General formula for the kk-th root’s argument: θk=θ+2πk5,k=0,1,2,3,4\theta_k = \frac{\theta + 2\pi k}{5}, \quad k = 0, 1, 2, 3, 4

Since θ=0\theta = 0, the arguments are: θk=2πk5\theta_k = \frac{2\pi k}{5}

Specifically:

• θ0=0\theta_0 = 0 radians
• θ1=2π5\theta_1 = \frac{2\pi}{5} radians
• θ2=4π5\theta_2 = \frac{4\pi}{5} radians
• θ3=6π5\theta_3 = \frac{6\pi}{5} radians
• θ4=8π5\theta_4 = \frac{8\pi}{5} radians

Step 4: Write the roots in polar or rectangular form
The fifth roots are: zk=3(cos⁡θk+isin⁡θk)z_k = 3 \left( \cos \theta_k + i \sin \theta_k \right)

Calculating each root explicitly:

1. z0=3(cos⁡0+isin⁡0)=3(1+0i)=3z_0 = 3(\cos 0 + i \sin 0) = 3(1 + 0i) = 3
2. z1=3(cos⁡2π5+isin⁡2π5)z_1 = 3\left(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\right)
3. z2=3(cos⁡4π5+isin⁡4π5)z_2 = 3\left(\cos \frac{4\pi}{5} + i \sin \frac{4\pi}{5}\right)
4. z3=3(cos⁡6π5+isin⁡6π5)z_3 = 3\left(\cos \frac{6\pi}{5} + i \sin \frac{6\pi}{5}\right)
5. z4=3(cos⁡8π5+isin⁡8π5)z_4 = 3\left(\cos \frac{8\pi}{5} + i \sin \frac{8\pi}{5}\right)

Using approximate values for cosine and sine:

• cos⁡2π5≈0.3090\cos \frac{2\pi}{5} \approx 0.3090, sin⁡2π5≈0.9511\sin \frac{2\pi}{5} \approx 0.9511
• cos⁡4π5≈−0.8090\cos \frac{4\pi}{5} \approx -0.8090, sin⁡4π5≈0.5878\sin \frac{4\pi}{5} \approx 0.5878
• cos⁡6π5≈−0.8090\cos \frac{6\pi}{5} \approx -0.8090, sin⁡6π5≈−0.5878\sin \frac{6\pi}{5} \approx -0.5878
• cos⁡8π5≈0.3090\cos \frac{8\pi}{5} \approx 0.3090, sin⁡8π5≈−0.9511\sin \frac{8\pi}{5} \approx -0.9511

So the roots in rectangular form:

1. z0=3+0iz_0 = 3 + 0i
2. z1=3(0.3090+0.9511i)=0.927+2.853iz_1 = 3(0.3090 + 0.9511i) = 0.927 + 2.853i
3. z2=3(−0.8090+0.5878i)=−2.427+1.763iz_2 = 3(-0.8090 + 0.5878i) = -2.427 + 1.763i
4. z3=3(−0.8090−0.5878i)=−2.427−1.763iz_3 = 3(-0.8090 – 0.5878i) = -2.427 – 1.763i
5. z4=3(0.3090−0.9511i)=0.927−2.853iz_4 = 3(0.3090 – 0.9511i) = 0.927 – 2.853i

Explanation
Finding the fifth roots of a positive real number like 243 involves expressing it in polar form and then applying De Moivre’s theorem. The magnitude of each root is the fifth root of the original magnitude. The arguments of the roots are evenly spaced by dividing the full circle (360 degrees or 2π2\pi radians) by the root degree, which is five in this case. The roots form a regular pentagon when plotted in the complex plane, all with the same magnitude but different angles. This approach generalizes to any complex number, where the polar form is essential to find all distinct roots.