Answer to Question #96508 in Complex Analysis for Olajide Olaitan

Question #96508
Find the fifth root of -1
1
Expert's answer
2019-10-22T06:55:01-0400

1) Let "a<0" and "a" is not a complex number.

If "n" is an odd number, then there is one and only one real number "x" such that "x^n = a". This number is "x = \\sqrt[n]{a}" and is called the root of an odd degree "n" from a negative number "a".

So, "\\sqrt[5]{-1} = \\sqrt[5]{(-1)*(-1)*(-1)*(-1)*(-1)}=\\sqrt[5]{(-1)^5} = -1."


2) Let "a \\in\\Complex" (complex number) and "a=i^2"

Find the trigonometric form of a complex number

"x=Re(a)=-1, y=Im(a)=0".

"x<0,y\\geq0\\implies arg(a)=\\phi=\\pi-\\arctan(\\cfrac{y}{\\vert x \\vert})=\\pi -0=\\pi".

Thus, the trigonometric form of the complex number "a=i^2" is "a=\\cos(\\pi)+i\\sin(\\pi)".

The fifth roots are "a_k = \\sqrt[5]{a}=\\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi k}{5}+i\\sin\\cfrac{\\phi+2\\pi k}{5} ), k=0,1,2,3,4".

"k=0\\implies a_0 = \\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi *0}{5}+i\\sin\\cfrac{\\phi+2\\pi *0}{5} )=\\cos\\cfrac{\\pi}{5}+i\\sin\\cfrac{\\pi}{5}=cos36\\degree+i\\sin36\\degree = 0.809017+0.587785i" ,

"k=1\\implies a_1 = \\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi *1}{5}+i\\sin\\cfrac{\\phi+2\\pi *1}{5} )=\\cos\\cfrac{3\\pi}{5}+i\\sin\\cfrac{3\\pi}{5}=cos108\\degree+i\\sin108\\degree=-0.309017+0.951057i" ,

"k=2\\implies a_2 = \\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi *2}{5}+i\\sin\\cfrac{\\phi+2\\pi *2}{5} )=\\cos\\cfrac{5\\pi}{5}+i\\sin\\cfrac{5\\pi}{5}=\\cos\\pi +i\\sin\\pi = -1" ,

"k=3\\implies a_3 = \\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi *3}{5}+i\\sin\\cfrac{\\phi+2\\pi *3}{5} )=\\cos\\cfrac{7\\pi}{5}+i\\sin\\cfrac{7\\pi}{5}=cos252\\degree+i\\sin252\\degree=-0.309017-0.951057i" ,

"k=4\\implies a_4 = \\sqrt[5]{\\vert a \\vert}(\\cos\\cfrac{\\phi+2\\pi *4}{5}+i\\sin\\cfrac{\\phi+2\\pi *4}{5} )=\\cos\\cfrac{9\\pi}{5}+i\\sin\\cfrac{9\\pi}{5}=cos324\\degree+i\\sin324\\degree=0.809017-0.587785i"

For evaluation was used cosine and sine tables (https://onlinemschool.com/math/formula/cosine_table/ , https://onlinemschool.com/math/formula/sine_table/ ).


Answer: "0.809017\\pm0.587785i, -0.309017\\pm0.951057i, -1."



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