Question

What is (-i)^5 ? a. -i b. 1 c. -1 d. i

Answer 1

`\huge\boxed{(-i)^5=-i}`

Explanation:

`√(-1)=i\to i^2=\left(√(-1)\right)^2=-1\\\\a^n\cdot a^m=a^(n+m)\\\\(ab)^n=a^nb^n\\=========================\\\\\text{We have:}\\\\(-i)^5=(-i)^(2+2+1)=(-i)^2(-i)^2(-i)^1=(-1\cdot i)^2(-1\cdot i)^2(-i)\\\\=(-1)^2(i)^2(-1)^2(i)^2(-i)=(1)(-1)(1)(-1)(-i)=-i`

Answer 2

Answer: Choice A. -i

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Work Shown:

(-i)^5 = (-1*i)^5

(-i)^5 = (-1)^5 * i^5

(-i)^5 = (-1)^5 * i^2*i^2*i

(-i)^5 = (-1)^5 * (-1)*(-1)*i

(-i)^5 = -1 * 1 * i

(-i)^5 = -i

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A shortcut to quickly computing i^5 is to note the remainder of 5/4 is 1, so this means that i^5 = i^1 = i. Another example is i^25 would equal the same thing since 25/4 has a remainder of 1 as well.