Answered: The powers of the imaginary unit are…

Name: The powers of the imaginary unit are cyclic of order 4. The first four
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Description: The powers of the imaginary unit are cyclic of order 4. The first four powers of I area: i^0= 1,i^1=i,i^2=−1, and i^3=−i. Since the powers are cyclic we have the relationshipin=i^n−4 for n≥4. This relationship can be used to calculate the powers of i

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The powers of the imaginary unit are cyclic of order 4. The first four powers of I area: i^0= 1,i^1=i,i^2=−1, and i^3=−i. Since the powers are cyclic we have the relationshipin=i^n−4 for n≥4. This relationship can be used to calculate the powers of i. Calculate i^729642

Expert Solution

Step 1

We have to calculate:

$i^{729642}$

We have

$\begin{array}{rcl} i^{0} & = & 1 \\ i^{1} & = & i \\ i^{2} & = & - 1 \\ i^{3} & = & - i \\ i^{4} & = & 1 \end{array}$

Therefore imaginary unit $i$ is completing their cycle at power four

which means if power of imaginary unit $i$ is multiple of four then the whole expression would be equal to one.

$i^{4 n} = 1$ where $n$ is positive integer

Thus dividing the power of imaginary unit $i$ by $4$ ,

$\frac{729642}{4} = 182410 (r e m a i n d e r = 2)$

using Euclid division lemma,

$\begin{array}{rcl} a & = & b q + r \\ 729642 & = & 182410 \times 4 + 2 \end{array}$

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