Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+12*x=0
-
Equation x^3=4x
-
Equation (x-9)^3+3*(x-9)^2=0
-
Equation (x+3)^4-4(x+3)^2-5=0
- Express {x} in terms of y where:
- 15*x+2*y=8
- 1*x+4*y=-11
- 5*x-18*y=-17
- -9*x-8*y=-20
- Factor polynomial:
- x^8+1
-
Identical expressions
- x^ eight + one = zero
- x to the power of 8 plus 1 equally 0
- x to the power of eight plus one equally zero
- x8+1=0
- x⁸+1=0
- x^8+1=O
-
Similar expressions
- x^8-1=0
x^8+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{8} + 1 = 0$$
Because equation degree is equal to = 8 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 8 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{8} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{8} e^{8 i p} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{8 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = -1$$
so
$$\cos{\left(8 p \right)} = -1$$
and
$$\sin{\left(8 p \right)} = 0$$
then
$$p = \frac{\pi N}{4} + \frac{\pi}{8}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$x_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$x^{8} + 1 = 0$$
Because equation degree is equal to = 8 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 8 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{8} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{8} e^{8 i p} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{8 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = -1$$
so
$$\cos{\left(8 p \right)} = -1$$
and
$$\sin{\left(8 p \right)} = 0$$
then
$$p = \frac{\pi N}{4} + \frac{\pi}{8}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$x_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
Rapid solution
[src]
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
x1 = - / - - ----- + I* / - + -----
\/ 2 4 \/ 2 4
$$x_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
x2 = / - - ----- - I* / - + -----
\/ 2 4 \/ 2 4
$$x_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
x3 = - / - + ----- - I* / - - -----
\/ 2 4 \/ 2 4
$$x_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
x4 = / - + ----- + I* / - - -----
\/ 2 4 \/ 2 4
$$x_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
/ ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___
| ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2
|\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - + ----- \/ 2 * / - - -----
| \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4
x5 = I*|---------------------- + ----------------------| + ---------------------- - ----------------------
\ 2 2 / 2 2
$$x_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
/ ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___
| ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2
|\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + -----
| \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4
x6 = I*|---------------------- - ----------------------| + ---------------------- + ----------------------
\ 2 2 / 2 2
$$x_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$
/ ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___
| ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2
|\/ 2 * / - + ----- \/ 2 * / - - ----- | \/ 2 * / - - ----- \/ 2 * / - + -----
| \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4
x7 = I*|---------------------- - ----------------------| - ---------------------- - ----------------------
\ 2 2 / 2 2
$$x_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
/ ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___
| ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2
| \/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + -----
| \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4
x8 = I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------
\ 2 2 / 2 2
$$x_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$
x8 = -sqrt(2)*sqrt(sqrt(2)/4 + 1/2)/2 + sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*(-sqrt(2)*sqrt(sqrt(2)/4 + 1/2)/2 - sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2)
Sum and product of roots
[src]
sum
/ ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2
/ ___ / ___ / ___ / ___ / ___ / ___ / ___ / ___ |\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - + ----- \/ 2 * / - - ----- |\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- |\/ 2 * / - + ----- \/ 2 * / - - ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + -----
/ 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4
- / - - ----- + I* / - + ----- + / - - ----- - I* / - + ----- + - / - + ----- - I* / - - ----- + / - + ----- + I* / - - ----- + I*|---------------------- + ----------------------| + ---------------------- - ---------------------- + I*|---------------------- - ----------------------| + ---------------------- + ---------------------- + I*|---------------------- - ----------------------| - ---------------------- - ---------------------- + I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------
\/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)$$
=
/ ___________ ___________\ / ___________ ___________\ / ___________ ___________\ / ___________ ___________\ | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | ___ / 1 \/ 2 ___ / 1 \/ 2 | | ___ / 1 \/ 2 ___ / 1 \/ 2 | | ___ / 1 \/ 2 ___ / 1 \/ 2 | | ___ / 1 \/ 2 ___ / 1 \/ 2 | |\/ 2 * / - - ----- \/ 2 * / - + ----- | |\/ 2 * / - - ----- \/ 2 * / - + ----- | |\/ 2 * / - + ----- \/ 2 * / - - ----- | | \/ 2 * / - - ----- \/ 2 * / - + ----- | | \/ 2 4 \/ 2 4 | | \/ 2 4 \/ 2 4 | | \/ 2 4 \/ 2 4 | | \/ 2 4 \/ 2 4 | I*|---------------------- + ----------------------| + I*|---------------------- - ----------------------| + I*|---------------------- - ----------------------| + I*|- ---------------------- - ----------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
product
/ / ___________ ___________\ ___________ ___________\ / / ___________ ___________\ ___________ ___________\ / / ___________ ___________\ ___________ ___________\ / / ___________ ___________\ ___________ ___________\
| | / ___ / ___ | / ___ / ___ | | | / ___ / ___ | / ___ / ___ | | | / ___ / ___ | / ___ / ___ | | | / ___ / ___ | / ___ / ___ |
/ ___________ ___________\ / ___________ ___________\ / ___________ ___________\ / ___________ ___________\ | | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | | | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | | | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 | | | ___ / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 ___ / 1 \/ 2 |
| / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | |\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - + ----- \/ 2 * / - - ----- | | |\/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- | | |\/ 2 * / - + ----- \/ 2 * / - - ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- | | | \/ 2 * / - - ----- \/ 2 * / - + ----- | \/ 2 * / - - ----- \/ 2 * / - + ----- |
| / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | | | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | | | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | | | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | | | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 |
|- / - - ----- + I* / - + ----- |*| / - - ----- - I* / - + ----- |*|- / - + ----- - I* / - - ----- |*| / - + ----- + I* / - - ----- |*|I*|---------------------- + ----------------------| + ---------------------- - ----------------------|*|I*|---------------------- - ----------------------| + ---------------------- + ----------------------|*|I*|---------------------- - ----------------------| - ---------------------- - ----------------------|*|I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------|
\ \/ 2 4 \/ 2 4 / \\/ 2 4 \/ 2 4 / \ \/ 2 4 \/ 2 4 / \\/ 2 4 \/ 2 4 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$
=
1
$$1$$
1
Numerical answer
[src]
x1 = -0.38268343236509 + 0.923879532511287*i
x2 = -0.38268343236509 - 0.923879532511287*i
x3 = -0.923879532511287 - 0.38268343236509*i
x4 = 0.923879532511287 - 0.38268343236509*i
x5 = -0.923879532511287 + 0.38268343236509*i
x6 = 0.38268343236509 - 0.923879532511287*i
x7 = 0.38268343236509 + 0.923879532511287*i
x8 = 0.923879532511287 + 0.38268343236509*i
x8 = 0.923879532511287 + 0.38268343236509*i
The graph