Answer to Question #106340 in Complex Analysis for Nikesh

Question #106340

Obtain the 6th roots of -7 , and represent them in an Argand diagram.


1
Expert's answer
2020-03-24T10:04:14-0400

"z=-7"

"x=Re(z)=-7"

"y=Im(z)=0"

"|z|=\\sqrt{x^2+y^2}=7"

"\\phi =arg(z)=\\pi-arctan(y\/|x|)=\\pi-arctan(0\/7)=\\pi-0=\\pi"

"z=7(cos\\pi+isin\\pi)"

To find the 6th roots we use this formula

"z_k=\\sqrt[6]{z}=\\sqrt[6]{|z|}(cos {\\frac {\\phi+2\\pi k} 6}+isin{\\frac {\\phi+2\\pi k} 6}), k=0,1,2,3,4,5"

So, let's find all "z_k"

"z_0=\\sqrt[6]{7}(cos {\\frac {\\pi} 6}+isin{\\frac {\\pi} 6})"

"z_1=\\sqrt[6]{7}(cos {\\frac {\\pi} 2}+isin{\\frac {\\pi} 2})"

"z_2=\\sqrt[6]{7}(cos {\\frac {5\\pi} 6}+isin{\\frac {5\\pi} 6})"

"z_3=\\sqrt[6]{7}(cos {\\frac {7\\pi} 6}+isin{\\frac {7\\pi} 6})"

"z_4=\\sqrt[6]{7}(cos {\\frac {3\\pi} 2}+isin{\\frac {3\\pi} 2})"

"z_5=\\sqrt[6]{7}(cos {\\frac {11\\pi} 6}+isin{\\frac {11\\pi} 6})"

Now, let's plot all roots on Argand diagram


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS