The fourth root of the complex number will be,[tex]\alpha=-1.01 +0.65028\ i[/tex]
A complex number is one that has both a real and an imaginary component, both of which are preceded by the letter I which stands for the square root of -1.
The given complex number as;
[tex]\rm Z=1+\sqrt{3} \ i[/tex]
The complex number in the polar form as;
[tex]\rm r= \sqrt{1^2+(\sqrt3)^2} \\\\ r=\sqrt{10} \\\\ r=3.16[/tex]
[tex]\rm \tan \theta = \frac{\sqrt3}{1} \\\\ \theta = 60^0[/tex]
The fourth root of the complex number is found as;
[tex]\rm \alpha = \sqrt[4]{r} (cos\frac{\theta}{4} )+isin(\frac{\theta}{4} )\\\\ \rm \alpha = \sqrt[4]{3.16} (cos\frac{60}{4} )+i \ sin(\frac{60}{4} )\\\\ \alpha=1.33 \times -0.7596+0.65028 \ i \\\\\ \alpha=-1.01 +0.65028\ i[/tex]
Hence, the fourth root of the complex number will be,[tex]\alpha=-1.01 +0.65028\ i[/tex]
To learn more about the complex number, refer to the link;
https://brainly.com/question/10251853
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