document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165887>Question 1165887</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Express the roots of (-14+3i)^-2/5 complex number in polar form. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852878 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "# W = -14 + 3i<BR>\n" );
document.write( "# Modulus r = sqrt(14^2 + 3^2) = sqrt(205)<BR>\n" );
document.write( "r_squared = 205<BR>\n" );
document.write( "R = 1 / r_squared  # Modulus of C = W^-2\r<BR>\n" );
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document.write( "# Argument calculation for W<BR>\n" );
document.write( "alpha = np.arctan(3/14) # Reference angle<BR>\n" );
document.write( "# phi_W = pi - alpha (Argument of W)\r<BR>\n" );
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document.write( "# Argument calculation for C = W^-2<BR>\n" );
document.write( "# phi = -2 * phi_W = -2 * (pi - alpha) = -2*pi + 2*alpha. We use the principal argument 2*alpha.<BR>\n" );
document.write( "phi = 2 * alpha\r<BR>\n" );
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document.write( "# Modulus of the roots (R_root)<BR>\n" );
document.write( "R_root = R**(1/5)<BR>\n" );
document.write( "R_root_decimal = R_root # (1/205)^(1/5)\r<BR>\n" );
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document.write( "# Calculate phi in degrees for understanding<BR>\n" );
document.write( "phi_degrees = np.degrees(phi)\r<BR>\n" );
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document.write( "# Calculate the argument for k=0<BR>\n" );
document.write( "theta_0 = phi / 5\r<BR>\n" );
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document.write( "# Calculate the arguments for k=0, 1, 2, 3, 4 (in radians)<BR>\n" );
document.write( "k_values = np.arange(5)<BR>\n" );
document.write( "theta_k = (phi + 2 * np.pi * k_values) / 5\r<BR>\n" );
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document.write( "print(f\"Modulus of the roots (R_root): {R_root_decimal}\")<BR>\n" );
document.write( "print(f\"Base angle phi (radians): {phi}\")<BR>\n" );
document.write( "print(f\"Argument theta_k (radians): {theta_k}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Modulus of the roots (R_root): 0.34486508519906256<BR>\n" );
document.write( "Base angle phi (radians): 0.4221866664454931<BR>\n" );
document.write( "Argument theta_k (radians): [0.08443733 1.34107439 2.59771146 3.85434852 5.11098558]\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The problem asks for the roots of the complex number $Z = (-14+3i)^{-2/5}$ in polar form. We interpret this as finding the **five fifth roots** of the complex number $C = (-14+3i)^{-2}$.\r<BR>\n" );
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document.write( "The polar form of a complex number is $Z = R e^{i\theta}$. The $n$-th roots of $Z$ are given by De Moivre's Theorem:<BR>\n" );
document.write( "$$Z_k = \sqrt[n]{R} \cdot e^{i \left(\frac{\theta + 2\pi k}{n}\right)}, \quad \text{for } k=0, 1, 2, \dots, n-1$$\r<BR>\n" );
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document.write( "Here, $n=5$.\r<BR>\n" );
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document.write( "## 1\. Convert the Base Complex Number to Polar Form\r<BR>\n" );
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document.write( "Let $W = -14 + 3i$. We first convert $W$ to polar form $W = r e^{i\theta}$.<BR>\n" );
document.write( "$$r = \sqrt{(-14)^2 + 3^2} = \sqrt{196 + 9} = \sqrt{205}$$<BR>\n" );
document.write( "$$\theta = \pi - \tan^{-1}\left(\frac{3}{14}\right) \quad (\text{Since } W \text{ is in QII})$$\r<BR>\n" );
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document.write( "Now we find $C = W^{-2}$:\r<BR>\n" );
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document.write( "  * **Modulus ($R$):**<BR>\n" );
document.write( "    $$R = r^{-2} = (\sqrt{205})^{-2} = \frac{1}{205}$$<BR>\n" );
document.write( "  * **Argument ($\phi$):**<BR>\n" );
document.write( "    $$\phi = -2\theta = -2\left(\pi - \tan^{-1}\left(\frac{3}{14}\right)\right)$$<BR>\n" );
document.write( "    Using the periodicity of the argument, we take the principal argument, $\phi = 2 \tan^{-1}\left(\frac{3}{14}\right)$.\r<BR>\n" );
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document.write( "$$\phi \approx 2 \times 0.211093 \approx 0.42219 \text{ radians}$$\r<BR>\n" );
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document.write( "Thus, $C \approx \frac{1}{205} e^{i(0.42219)}$.\r<BR>\n" );
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document.write( "## 2\. Find the Five Fifth Roots ($Z_k$)\r<BR>\n" );
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document.write( "The roots $Z_k$ are the five distinct values of $C^{1/5}$.\r<BR>\n" );
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document.write( "### A. Modulus of the Roots ($R_{\text{root}}$)\r<BR>\n" );
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document.write( "$$R_{\text{root}} = \sqrt[5]{R} = \sqrt[5]{\frac{1}{205}} \approx \mathbf{0.3449}$$\r<BR>\n" );
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document.write( "### B. Arguments of the Roots ($\theta_k$)\r<BR>\n" );
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document.write( "The arguments are calculated using the formula:<BR>\n" );
document.write( "$$\theta_k = \frac{\phi + 2\pi k}{5}, \quad \text{for } k=0, 1, 2, 3, 4$$\r<BR>\n" );
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document.write( "We use the exact value $\phi = 2 \tan^{-1}\left(\frac{3}{14}\right)$ and the approximation $\phi \approx 0.42219$.\r<BR>\n" );
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document.write( "| k | Argument $\theta_k$ (Exact Form) | Argument $\theta_k$ (Approx. Radians) |<BR>\n" );
document.write( "| :---: | :---: | :---: |<BR>\n" );
document.write( "| 0 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) \right)$ | $\mathbf{0.0844}$ |<BR>\n" );
document.write( "| 1 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 2\pi \right)$ | $\mathbf{1.3411}$ |<BR>\n" );
document.write( "| 2 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 4\pi \right)$ | $\mathbf{2.5977}$ |<BR>\n" );
document.write( "| 3 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 6\pi \right)$ | $\mathbf{3.8543}$ |<BR>\n" );
document.write( "| 4 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 8\pi \right)$ | $\mathbf{5.1110}$ |\r<BR>\n" );
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document.write( "The roots of the complex number $(-14+3i)^{-2/5}$ in polar form are:<BR>\n" );
document.write( "$$Z_k = \frac{1}{\sqrt[5]{205}} \cdot e^{i \left( \frac{2 \tan^{-1}(3/14) + 2\pi k}{5} \right)}, \quad \text{for } k=0, 1, 2, 3, 4$$ </SPAN>\n" );
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