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Which of the following is the correct form of $-3i^8 - 3i^6 - 3i^{-1}$ , where the imaginary number $i$ is such that $i^2 = -1$ ? $\newline$ (A) $-1$ $\newline$ (B) $-1 - 3i$ $\newline$ (C) $-7 - 3i$ $\newline$ (D) $-1 - 3i$

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Transformations of functions

Full solution

Q. Which of the following is the correct form of

-3i^8 - 3i^6 - 3i^{-1}

, where the imaginary number

i

is such that

i^2 = -1

?

\newline

(A)

-1

\newline

(B)

-1 - 3i

\newline

(C)

-7 - 3i

\newline

(D)

-1 - 3i

Simplify $-3i^8$ : We need to simplify each term involving $i$ by using the fact that $i^2 = −1$ . Let's start with the term $−3i^8$ . Since $i^4 = (i^2)^2 = (−1)^2 = 1$ , we can simplify $i^8$ as $(i^4)^2 = 1^2 = 1$ . Therefore, $−3i^8$ simplifies to $−3 \times 1 = −3$ .
Simplify $-3i^6$ : Now let's simplify the term $-3i^6$ . Since $i^6 = (i^4)(i^2) = 1 \times (-1) = -1$ , we can simplify $-3i^6$ as $-3 \times (-1) = 3$ .
Simplify $-3i$ : The term $−3i$ does not simplify further since it's already in terms of $i$ to the first power. So, $−3i$ remains as it is.
Simplify $-1$ : The last term is $−1$ , which also does not simplify further.
Combine simplified terms: Now we combine all the simplified terms: $-3$ (from $-3i^8$ ) + $3$ (from $-3i^6$ ) $- 3i - 1$ . This simplifies to $(-3 + 3) - 3i - 1$ .
Final simplification: Combining like terms, we get $0 - 3i - 1$ , which simplifies to $-3i - 1$ .

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