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Home » Convert to exponential form: \( 8x = 2^5 \). - Dachbleche24

Convert to exponential form: \( 8x = 2^5 \). - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 eq b $ implies $ heta

📌 eq \phi $, so $ S $ is purely imaginary. However, the original question may expect a real value. Re-examining:

📌 Let $ S = rac{(a + b)^2 + (a - b)^2}{a^2 - b^2} = rac{2a^2 + 2b^2}{a^2 - b^2} $. Let $ a = 1 $, $ b = i $:

📌 S = rac{2(1 + (-1))}{1 - (-1)} = 0, ext{ but } a^2 - b^2 = 1 - (-1) = 2, ext{ numerator } 2(1 + (-1)) = 0 \Rightarrow S = 0.

📌 Another example: $ a = 1 $, $ b = e^{i2\pi/3} $:

📌 a^2 = 1, b^2 = e^{i4\pi/3}, a^2 - b^2 = 1 - e^{i4\pi/3}, a^2 + b^2 = 1 + e^{i4\pi/3}.

📌 Numerator: $ 2(1 + e^{i4\pi/3}) $, denominator: $ 1 - e^{i4\pi/3} $, so $ S = 2 \cdot rac{1 + e^{i4\pi/3}}{1 - e^{i4\pi/3}} $. Multiply numerator and denominator by $ e^{-i2\pi/3} $:

📌 S = 2i \cdot rac{e^{i2\pi/3} + e^{i2\pi/3}}{e^{i2\pi/3} - e^{-i2\pi/3}} = ext{complicated}.

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