The possible values of a + b are 0.89, 3.01 and -3.9
The given parameters are:
y = a + ib
z = 24 + i3
Where:
y = ∛z
Take the cube of both sides
y³ = z
Substitute the values for y and z
(a + ib)³ = 24 + i3
Expand
a³ + 3a²(ib) + 3a(ib)² + (ib)³ = 24 + i3
Further, expand
a³ + i3a²b + i²3ab² + i³b³ = 24 + i3
In complex numbers;
i² = -1 and i³= -i
So, we have:
a³ + i3a²b + (-1)3ab² -ib³ = 24 + i3
Further expand
a³ + i3a²b - 3ab² - ib³ = 24 + i3
By comparing both sides of the equation, we have:
a³ - 3ab² = 24
i3a²b - ib³ = i3
Divide through by i
3a²b - b³ = 3
So, we have:
a³ - 3ab² = 24
3a²b - b³ = 3
Using a graphing tool, we have:
(a,b) = (-1.55, 2.44), (2.89,0.12) and (-1.34,-2.56)
Add these values
a + b = 0.89, 3.01 and -3.9
Hence, the possible values of a + b are 0.89, 3.01 and -3.9
Read more about complex numbers at:
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