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=7z=-7

x=Re(z)=7x=Re(z)=-7

y=Im(z)=0y=Im(z)=0

z=x2+y2=7|z|=\sqrt{x^2+y^2}=7

ϕ=arg(z)=πarctan(y/x)=πarctan(0/7)=π0=π\phi =arg(z)=\pi-arctan(y/|x|)=\pi-arctan(0/7)=\pi-0=\pi

z=7(cosπ+isinπ)z=7(cos\pi+isin\pi)

To find the 6th roots we use this formula

zk=z6=z6(cosϕ+2πk6+isinϕ+2πk6),k=0,1,2,3,4,5z_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 zkz_k

z0=76(cosπ6+isinπ6)z_0=\sqrt[6]{7}(cos {\frac {\pi} 6}+isin{\frac {\pi} 6})

z1=76(cosπ2+isinπ2)z_1=\sqrt[6]{7}(cos {\frac {\pi} 2}+isin{\frac {\pi} 2})

z2=76(cos5π6+isin5π6)z_2=\sqrt[6]{7}(cos {\frac {5\pi} 6}+isin{\frac {5\pi} 6})

z3=76(cos7π6+isin7π6)z_3=\sqrt[6]{7}(cos {\frac {7\pi} 6}+isin{\frac {7\pi} 6})

z4=76(cos3π2+isin3π2)z_4=\sqrt[6]{7}(cos {\frac {3\pi} 2}+isin{\frac {3\pi} 2})

z5=76(cos11π6+isin11π6)z_5=\sqrt[6]{7}(cos {\frac {11\pi} 6}+isin{\frac {11\pi} 6})

Now, let's plot all roots on Argand diagram


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