fractional powers

Powers and roots of complex numbers have a strong geometric flavor to them.

\(\blacktriangleright \) How do we calculate \((3 + 3\sqrt {3} \, i)^{\tfrac {5}{2}}\) ?

0.1 Trigonometric Thinking

Step 1: Convert the base to trigonometric form.

\[ 3 + 3\sqrt {3} \, i \]
\[ = 6 \left ( \frac {1}{2} + \frac {\sqrt {3}}{2} \, i \right ) \]
\[ = 6 \left (\cos \left ( \frac {\pi }{3} \right )+i \, \sin \left ( \frac {\pi }{3} \right ) \right ) \]

Step 2: The Angle

Find \(\frac {\pi }{3}\) on the unit circle

We need to move the angle around.

First, the \(\tfrac {1}{2}\) part of the exponent means we need \(\tfrac {1}{2}\) of the angle: \(\frac {\pi }{6}\).

Second, the \(5\) in the exponent means we need to multiply the angle by \(5\): \(\frac {5\pi }{6}\).

Step 3: The Modulus

The \(6\) also has the exponent. Here we think just in terms of real numbers: \(6^{\tfrac {5}{2}}\).

Step 4: Reassemble

\[ 6^{\tfrac {5}{2}} \left (\cos \left ( \frac {5\pi }{6} \right )+i \, \sin \left ( \frac {5\pi }{6} \right ) \right ) \]
\[ = 6^{\tfrac {5}{2}} \left ( -\frac {\sqrt {3}}{2} + \frac {1}{2} \, i \right ) \]

0.2 Exponential Thinking

Step 1: Convert the base to exponential form.

\[ 3 + 3\sqrt {3} \, i \]

\(\sqrt {3^2 + (3\sqrt {3})^2} = \sqrt {9 + 9 \cdot 3} = \sqrt {36} = 6\)

\(\arctan \left ( \frac {3\sqrt {3}}{3} \right ) = \arctan \left ( \sqrt {3} \right ) = \frac {\pi }{3}\)

\[ = 6 \cdot e^{\tfrac {\pi }{3} \, i} = e^{\ln (6) + \tfrac {\pi }{3} \, i} \]

Step 2: The Exponent

\[ = \left ( e^{\ln (6) + \tfrac {\pi }{3} \, i} \right )^{\tfrac {5}{2}} \]
\[ = e^{\tfrac {5}{2} (\ln (6) + \tfrac {\pi }{3} \, i)} \]
\[ = e^{\tfrac {5}{2} \ln (6) + \tfrac {5\pi }{6} \, i} \]
\[ = 6^{\tfrac {5}{2}} \cdot e^{\tfrac {5\pi }{6} \, i} \]

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more examples can be found by following this link
More Examples of Complex Exponentials

2025-01-07 03:54:55