Which of the following is a negative integer if i= $sqrt(-1)$ ? A) $i^24$ B) $i^33$ C) $i^46$ D) $i^55$ E) $i^72$

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Answer 1 · 2018-03-18T18:48:26

$i^46$

$i^1 = i$

$i^2 = sqrt(-1) * sqrt(-1) = -1$

$i^3 = -1 * i = -i$

$i^4 = (i^2)^2 = (-1)^2 = 1$

the powers of $i$ are $i, -1, -i, 1$ , continuing in a cyclical sequence every $4$ th power.

in this set, the only negative integer is $-1$ .

for the power of $i$ to be a negative integer, the number that $i$ is raised to must be $2$ more than a multiple of $4$ .

$44/4 = 11$

$46 = 44+2$

$i^46 = i^2 = -1$

Answer 2 · 2018-03-21T11:01:03

C) $i^46$

Note that:

$i^0 = 1$

$i^1 = i$

$i^2 = -1$

$i^3 = -i$

$i^4 = 1$

So the increasing powers of $i$ will follow a pattern conforming to:

$i^(4k) = 1$

$i^(4k+1) = i$

$i^(4k+2) = -1$

$i^(4k+3) = -i$

for any integer $k$

The only one of these values which is negative is $i^(4k+2) = -1$

Hence the correct answer is C) since $46 = 4*11 + 2$

Which of the following is a negative integer if i= sqrt(-1)sqrt(-1)? A) i^24i^24 B) i^33i^33 C) i^46i^46 D) i^55i^55 E) i^72i^72