you have (10-i) / (12 + 5i).
multiply both denominator and numerator by (12 - 5i).
you get:
(10 - i) * (12 - 5i)
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(12 - 5i) * (12 + 5i)
the denominator of (12 - 5i) * (12 + 5i) becomes:
12 * (12 + 5i) - 5i * (12 + 5i) through the distributive law of multiplication.
simplify this to get:
144 + 60i - 60i - 25i^2
the middle terms cancel out and you are left with:
144 - 25i^2
since i^2 = -1, then the denominator becomes:
144 + 25 which becomes 169
the numerator of (10 - i) * (12 - 5i) becomes:
10 * (12 - 5i) - i * (12 - 5i) through the distributive law of multiplication.
simplify this to get:
120 - 50i - 12i + 5i^2
combine like terms to get:
120 - 62i + 5i^2
since i^2 = -1, then the numerator becomes:
120 - 62i - 5 which becomes:
115 - 62i
your fraction becomes:
115 - 62i
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169
the rules for i are as follows:
i = sqrt(-1) = i
i^2 = sqrt(-1) * sqrt(-1) = -1
i^3 = -1 * sqrt(-1) = -i
i^4 = -1 * -1 = 1
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
i^9 is the same as i
i^10 is the same as i^2
i^11 is the same as i^3
i^12 is the same as i^4
what you do is take the exponent of the base i and divide it by 4 and the remainder is the value of i that you can then translate to the exponents of i from 1 to 4.
examples:
i^17 = i because the remainder after dividing by 17 by 4 is equal to 1.
i^18 = i^2 because the remainder after dividing 18 by 4 is equal to 2.
i^19 = i^3 because the remainder after dividing 19 by 4 is equal to 3.
i^20 = i^4 because the remainder after dividing 20 by 4 is equal to 0.