Compute the following quantities: a) i^21-i^32 b) 2-i/3-2i. Please show work

Question

Compute the following quantities: a) i^21-i^32

b) 2-i/3-2i.

Please show work

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Faith 2 weeks 2021-09-12T10:11:02+00:00 1 Answer 0

Answers ( )

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    2021-09-12T10:12:40+00:00

    Answer: a) i-1; b) 4+i/13

    Step-by-step explanation:

    The complex number i is defined as the number such that i^{2}=-1;

    We use the propperty to notice that i^1=i \quad i^2 = -1 \quad i^3=-i \quad i^4=1 \quad i^5=1 \quad i^6=-1 \quad i^7=-i \quad i^8=1 \quad i^9=i \quad i^10= -1 etc....

    a) We notice that i^{21}=i \quad \text{ and \text} \quad i^{32}=1. Hence, i^{21}-i^{32}=i-1.

    b) We multiply the expression by 1=\frac{3+2\cdot i}{3 + 2 \cdot i}. Then we get that

    \frac{2-i}{3-2 \cdot i}=\frac{2-i}{3-2\cdot i}\cdot\frac{3+2 \cdot i}{3 + 2\cdot i } = \frac{(2-i)\cdot(3+2 \cdot i)}{3^2+2^2}= \frac{6+4i-3i+2i^2}{13}=\frac{6+i-2}{13} = \frac{4+i}{13}

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