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How do you Simplify this

How do you Simplify this-example-1
User Gastaldi
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1 Answer

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First, recall some rules of exponents.


a^m * a^n = a^(m + n)


(a^m)/(a^n) = a^(m - n)


-a^n = - (a^n)


a^(-n) = (1)/(a^n)

Here is the solution.


(-i^(-44) \cdot i^(35))/(i^8) =


= -(i^(-9) )/(i^8)


= -i^(-17)


= -(1)/(i^(17))


= -(1)/(i)

(For an explanation of how
i^(17) became
i,
see below the final answer.)


= -(1)/(i) * (i)/(i)


= - (i)/(i^2)


= (-i)/(-1)


= i

Final answer:
i



Integer powers of i.


i^0 = 1


i^1 = i


i^2 = -1


i^3 = -1 * i = -i


i^4 = i^3 * i = -i * i = 1 Notice that
i^4 = i^0

Now the pattern is complete, and for each consecutive integer power, you get the same pattern of results. Let n be a positive integer multiple of 4.
Then,
i^n = 1
i^(n + 1) = i
i^(n + 2) = -1
i^(n + 3) = -i

After power n + 3, power n + 4 is again an multiple of 4, and you start again with 1. Notice that integer 17 is 16 + 1, so it is 1 more than a multiple of 4, so i^17 is the same as i^(n + 1) = i.
User Matt Gibson
by
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