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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đđźđđŹđđąđšđ§: Express z = (2+5i)/i(3-2i) in form a+bi and find the value of a and b.
đđšđ„đźđđąđšđ§:
z = (2+5i) / (i(3-2i))
Denominator: i(3-2i) = 3i - 2i^2 = 3i - 2(-1) = 3i + 2 = 2 + 3i
z = (2+5i) / (2 + 3i)
Multiply by conjugate: ((2+5i)(2 - 3i)) / ((2 + 3i)(2 - 3i))
Numerator: 2(2) + 2(-3i) + 5i(2) + 5i(-3i) = 4 - 6i + 10i - 15i^2 = 4 + 4i + 15 = 19 + 4i
Denominator: 2^2 - (3i)^2 = 4 - 9i^2 = 4 + 9 = 13
z = (19 + 4i) / 13 = 19/13 + (4/13)i
Therefore, a = 19/13 and b = 4/13.