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Complex Numbers Question

Complex Numbers Question-example-1
User Jdiaz
by
8.1k points

1 Answer

6 votes

Since
i = √(-1), it follows that
i^2 = -1,
i^3 = -i, and
i^4 = 1.

Then


i^5 = i^4* i = i \\\\ i^6 = i^4* i^2 = -1 \\\\ i^7 = i^4* i^3 = -i \\\\ i^8 = i^4 * i^4 = 1

and the pattern repeats. The value of
i^n boils down to finding the remainder upon dividing
n by 4.

We have


i^(11) = i^8* i^3 = -i \\\\ i^(33) = i^(32) * i = \left(i^4\right)^8* i = i \\\\ i^(257) = i^(256) * i = \left(i^4\right)^(64)* i = i

so that


13 - 6i^(11) + 2i^(33) - 5i^(257) = 13 + 6i + 2i - 5i = \boxed{13 + 3i}

and so the answer is C.

User Ryan How
by
7.9k points

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