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Question 588556: Simplify the equation 9 - 6i - (2 - 3i)^2.
i tried 9-6i-(2-3i)(2-3i)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you had the right idea but you didn't go far enough.
you needed to multiply (2-3i) * (2-3i) to get the result of that multiplication.
based on the law of distributive multiplication, the result of that multiplication would be as follows:
(2-3i) * (2-3i) equals:
2*(2-3i) - 3i*(2-3i) which equals:
4 - 6i - 6i + 9i^2
subtract that from 9 - 6i and you get:
9 - 6i - (4 - 6i - 6i + 9i^2)
remove parentheses to get:
9 - 6i - 4 + 6i + 6i - 9i^2
combine like terms to get:
5 + 6i - 9i^2
since i^2 = -1, this becomes:
5 + 6i - 9*(-1) which becomes:
5 + 6i + 9
combine like terms to get:
14 + 6i
that's your result.
the properties for imaginary numbers are:
i = square root of (-1)
i^2 = -1
i^3 = - square root of (-1)
i^4 = 1
this pattern then repeats.
i^5 is the same as i
i^6 is the same as i^2
i^7 is the same as i^3
i^8 is the same as i^4
etc.
if you want to know the value of i^7, you would do the following.
divide the exponent by 4 and then your equivalent expression will be i raised to the remainder of the division.
example:
i^7 is translated to its equivalent value as follows:
7/4 = 1 with a remainder of 3.
the base value of i^7 is equal to i^3.
the equivalent value of i^3 is equal to - square root of (-1).
all the calculations are done with i remaining as a variable and then when all the calculations are completed, you translate i to it's equivalent value.
in some cases, you can leave the i unchanged.
in other cases, you will need to translate the i to it's equivalent value.
it depends on the requirements of the problem and what your instructor expects you to do.
another example of converting the i to it's equivalent value:
i^37 is translated to it's equivalent value as follows:
37/4 = 9 with a remainder of 1.
the base value of i^37 is equal to i^1.
the equivalent value of i^1 is equal to square root of (-1).
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