What is the value of #i^ { - 343}#?

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Answer 1 · 2018-02-01T06:15:18

#i^-1#

For any #x#, when it's taken to a negative exponent, we can write:

#x^-1=1/x#

which gives us

#i^-343=1/i^343#

Moving to the large exponent. When working with #i#, it's important to remember this patterning:

#i^1=(sqrt(-1))^1=i#

#i^2=(sqrt(-1))^2=-1#

#i^3=(sqrt(-1))^3=-i#

#i^4=(sqrt(-1))^4=1#

From here, #i^5=i^4xxi=1xxi=i# and so on. So the question is - where does 343 sit in this pattern?

#343/4=85 3/4# - and so we can write it for our question as:

#1/(i^340xxi^3)=1/(1xx-i)=-1/i=i^-1#