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 (2+√-3)/(4+√-3) in form a+ib, a b ∈ R = √-1. State the value of a and b.
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
First, rewrite √-3 using i:
√-3 = √(3 * -1) = √3 * √-1 = √3 i
Now the expression becomes:
z = (2 + √3 i) / (4 + √3 i)
Multiply the numerator and denominator by the conjugate of the denominator (4 - √3 i):
z = ((2 + √3 i)(4 - √3 i)) / ((4 + √3 i)(4 - √3 i))
Expand the numerator:
(2 + √3 i)(4 - √3 i) = 2(4) + 2(-√3 i) + √3 i(4) + √3 i(-√3 i)
= 8 - 2√3 i + 4√3 i - 3 i^2
= 8 + 2√3 i - 3(-1)
= 8 + 2√3 i + 3
= 11 + 2√3 i
Expand the denominator:
(4 + √3 i)(4 - √3 i) = 4^2 - (√3 i)^2
= 16 - (3 * i^2)
= 16 - (3 * -1)
= 16 + 3
= 19
So, z = (11 + 2√3 i) / 19 = 11/19 + (2√3 / 19)i
Therefore, a = 11/19 and b = 2√3 / 19.