Level up your complex number skills! This video covers more challenging examples of expressing complex numbers in the standard form a + bi. We'll show you step-by-step how to combine multiplication, division, and other operations to simplify complex number expressions and achieve the a + bi form. Perfect for complex number practice!
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𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Express i(4+3i)/(1-i) in form a+ib, a b ∈ R = √-1. State the value of a and b.
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
First, multiply the numerator:
i(4+3i) = 4i + 3i^2 = 4i + 3(-1) = -3 + 4i
Now we have:
z = (-3 + 4i) / (1 - i)
Multiply the numerator and denominator by the conjugate of the denominator (1+i):
z = ((-3 + 4i)(1 + i)) / ((1 - i)(1 + i))
Expand the numerator:
(-3 + 4i)(1 + i) = -3(1) + -3(i) + 4i(1) + 4i(i)
= -3 - 3i + 4i + 4i^2
= -3 + i + 4(-1)
= -3 + i - 4
= -7 + i
Expand the denominator:
(1 - i)(1 + i) = 1^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2
So, z = (-7 + i) / 2 = -7/2 + (1/2)i
Therefore, a = -7/2 and b = 1/2.