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How do you find the 3rd root of #8e^(45i)#?

Answer 1

Roots are #2[cos (pi/12) +isin(pi/12)], 2[cos ((3pi)/4) +((isin(3pi))/4)] & 2[cos ((17pi)/12) +((isin(17pi))/12)]#

#8e^(45i)=8(cos45+isin45)=8(cos (pi/4)+isin(pi/4)):.# 1st root: #[8(cos pi/4)+(isinpi/4)]^(1/3)= 8^(1/3)[ cos (pi/(4*3)) +isin(pi/(4*3)]=2[cos (pi/12) +isin(pi/12)] :.#for 2nd root #(2pi)/3# to be added to the angle i.e #theta=(pi/12+(2pi)/3)=(9pi)/12=(3pi)/4:.#2nd root is #2[cos ((3pi)/4) +((isin(3pi))/4)]# for 3rd root #(2pi)/3# to be added to the angle i.e #theta=((3pi)/4+(2pi)/3)=(17pi)/12:.#3rd root is #2[cos ((17pi)/12) +((isin(17pi))/12)]#[Ans]

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Answer 2

Emery Adkins

To find the third root of (8e^{45i}), you can use De Moivre's theorem. First, express (8e^{45i}) in its polar form. Then, apply De Moivre's theorem by raising the modulus to the power of (1/3) and dividing the argument by (3). This will give you the third roots of the complex number. Finally, express each root in rectangular form if necessary.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.