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 (3+2i)/(2-5i) + (3-2i)/(2+5i) in form a+ib, a b ∈ R = √-1. State the value of a and b.
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
We need to express z = (3+2i)/(2-5i) + (3-2i)/(2+5i) in the form a + ib, where a and b are real numbers and i = √-1.
First, let's simplify the first fraction (3+2i)/(2-5i) by multiplying the numerator and denominator by the conjugate of the denominator, which is (2+5i):
(3+2i)/(2-5i) * (2+5i)/(2+5i) = (3(2) + 3(5i) + 2i(2) + 2i(5i)) / (2² - (5i)²)
= (6 + 15i + 4i + 10i²) / (4 - 25i²)
Recall that i² = -1. Substitute this into the expression:
= (6 + 19i + 10(-1)) / (4 - 25(-1))
= (6 + 19i - 10) / (4 + 25)
= (-4 + 19i) / 29
= -4/29 + (19/29)i
Next, let's simplify the second fraction (3-2i)/(2+5i) by multiplying the numerator and denominator by the conjugate of the denominator, which is (2-5i):
(3-2i)/(2+5i) * (2-5i)/(2-5i) = (3(2) + 3(-5i) - 2i(2) - 2i(-5i)) / (2² - (5i)²)
= (6 - 15i - 4i + 10i²) / (4 - 25i²)
Recall that i² = -1. Substitute this into the expression:
= (6 - 19i + 10(-1)) / (4 - 25(-1))
= (6 - 19i - 10) / (4 + 25)
= (-4 - 19i) / 29
= -4/29 - (19/29)i
Now, let's add the two simplified fractions:
z = (-4/29 + (19/29)i) + (-4/29 - (19/29)i)
Combine the real parts:
-4/29 + (-4/29) = -8/29
Combine the imaginary parts:
(19/29)i + (-19/29)i = 0i
So, z = -8/29 + 0i
Therefore, the complex number is in the form a + ib where a = -8/29 and b = 0.