Learn how to express complex numbers in the standard form a + bi when division is involved! This video covers three examples, showing you step-by-step how to simplify complex number expressions with division and achieve the a + bi form. Perfect for complex number practice!
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𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Express z = i(9+6i)/(2-i) in form a+bi and find the value of a and b.
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
z = (i(9+6i)) / (2-i)
Numerator: i(9+6i) = 9i + 6i^2 = 9i + 6(-1) = -6 + 9i
z = (-6 + 9i) / (2 - i)
Multiply by conjugate: ((-6 + 9i)(2 + i)) / ((2 - i)(2 + i))
Numerator: -6(2) + -6(i) + 9i(2) + 9i(i) = -12 - 6i + 18i + 9i^2 = -12 + 12i - 9 = -21 + 12i
Denominator: 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5
z = (-21 + 12i) / 5 = -21/5 + (12/5)i
Therefore, a = -21/5 and b = 12/5.