Concept Byte: Powers of Complex Numbers

EXTENSION

You can use the rules for multiplying complex numbers to find powers of complex numbers.

Example 1

Compute and graph ( 2 i ) n , open 2 i , close to the n , comma  for n = 0 , 1 , 2 ,  and  3 . n equals 0 comma 1 comma 2 comma , and , 3 .

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Example 2

Compute and graph ( 2 − 3 i ) n , open 2 minus 3 i , close to the n , comma  for n = 0 , 1 , 2 ,  and  3 . n equals 0 comma 1 comma 2 comma , and , 3 .

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Exercises

1. Based on the graph in Example 1, predict the location of ( 2 i ) 5 . open 2 i , close to the fifth , .
2. Compute and graph ( − 3 i ) n open negative 3 i , close to the n  for n = 0 , 1 , 2 ,  and  3 . n equals 0 comma 1 comma 2 comma , and , 3 .
3. 1. Connect the points from the graph in Example 1 with a smooth curve. Estimate ( 2 i ) 1 2 . open , 2 i , close super 1 half end super . .
   2. Use a graphing calculator to compute ( 2 i ) 1 2 . open , 2 i , close super 1 half end super . .  Does it fall on the curve? Was it close to your estimate?
4. Use a graphing calculator to find values of ( 2 − 3 i ) n open 2 minus 3 i , close to the n  for n = 0 . 5 , 1 . 5 , n equals 0 . 5 comma 1 . 5 comma  and 2.5. Copy the graph and add these points.
5. Compute and graph ( 3 − 4 i ) n open 3 minus 4 i , close to the n  for n = 0 , 1 , 2 ,  and  3 . n equals 0 comma 1 comma 2 comma , and , 3 .

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