My work with complex numbers verified that the only possible

William Collins 2022-01-17 Answered
My work with complex numbers verified that the only possible cube root of 8 is 2. Determine whether the statement makes sense or does not make sense, and explain your reasoning.
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Expert Answer

temzej9
Answered 2022-01-18 Author has 30 answers

Step 1
Concept:
The calculus helps in understanding the changes between values that are related by a function. It is used in different areas such as physics, biology and sociology and so on. The calculus is the language of engineers, scientists and economics.
Step 2
Given:
It is given that,''My work with complex numbers verified that the only possible cube root of 8 is 2''
The objective is to check that the sentence make sense or not
Step 3
This given statement does not makes sense
Here DeMoivre's theorem for roots applies to complex numbers in polar form. Thus writing the given term in terms of polar form that is 8 or 8+0i in the polar form Expressing θ in radians although degrees could also be used
8=r(cosθ+isinθ)=8(cos0+isin0)
There is three cube root exactly of 8. From DeMoivre's theorem for finding complex roots, the cube roots of 8 are
Zk=83(cos(0+2πk3)+isin(0+2πk3))
k=0, 1, 2
Step 4
The cube roots of 8 are found by substituting 0, 1, 2 for k in the above expression
Z0=83(cos(2π03)+isin(2π03))
Z0=2(cos00+isin00) Polar form
Z0=2 rectangular form
Z1=2(cos(2π3)+isin(2π3))
Z1=2(cos120+isin120)
Thus polar form,
Z1=2(cos120+isin120)
The rectangular form
Z1=2(12+i32)
Step 5
Z2=83(cos(4π3)isin(4π3))
Z2=2(cos240+isin240)
polar form
Z2=2(cos60isin60)
Rectangular form
Z2=2(32+i12)

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