To find: The roots of the fifths root of 32 and sketch the roots in the complex plane.
The roots of fifth root of 32 are where .
Roots of a complex number:
Let and n be a positive integer. Then has the distinct nth roots where .
Rewrite the complex number 32 in polar form.
The polar form of the complex number is where and .
Consider the complex number 32.
Obtain the argument of the complex number 32.
Thus, the argument of the complex number 32 is
Obtain the modulus of the complex number 32.
Thus, the value of .
Therefore, the polar form of the complex number 32 is .
By the above theorem, the roots of eighth root of 32 are where .
Use online calculator to sketch the roots in the complex plane as shown below in Figure 1.
From figure 1, it is observed that all fifths roots of 32 form a polygon on complex plane.
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