Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation (a-2)(b+5)=(a-2)b+(a-2)5
-
Equation x+5=0
-
Equation 4*x^2-25=0
-
Equation 8x-15=-12+5x
- Express {x} in terms of y where:
- 8*x-6*y=-18
- -19*x-17*y=6
- -14*x-3*y=-14
- 9*x+9*y=11
- Factor polynomial:
- x^8-1
-
Identical expressions
- x^ eight - one = zero
- x to the power of 8 minus 1 equally 0
- x to the power of eight minus 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 - contains the even number 8 in the numerator, then
the equation has two real roots.
Get the root 8-th degree of the equation sides:
We get:
$$\sqrt[8]{x^{8}} = \sqrt[8]{1}$$
$$\sqrt[8]{x^{8}} = \left(-1\right) \sqrt[8]{1}$$
or
$$x = 1$$
$$x = -1$$
We get the answer: x = 1
We get the answer: x = -1
or
$$x_{1} = -1$$
$$x_{2} = 1$$
All other 6 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
$$p = \frac{\pi N}{4}$$
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:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - i$$
$$z_{4} = i$$
$$z_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{3} = - i$$
$$x_{4} = i$$
$$x_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x^{8} - 1 = 0$$
Because equation degree is equal to = 8 - contains the even number 8 in the numerator, then
the equation has two real roots.
Get the root 8-th degree of the equation sides:
We get:
$$\sqrt[8]{x^{8}} = \sqrt[8]{1}$$
$$\sqrt[8]{x^{8}} = \left(-1\right) \sqrt[8]{1}$$
or
$$x = 1$$
$$x = -1$$
We get the answer: x = 1
We get the answer: x = -1
or
$$x_{1} = -1$$
$$x_{2} = 1$$
All other 6 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
$$p = \frac{\pi N}{4}$$
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:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - i$$
$$z_{4} = i$$
$$z_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{3} = - i$$
$$x_{4} = i$$
$$x_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ ___ ___ ___ ___ ___ ___
\/ 2 I*\/ 2 \/ 2 I*\/ 2 \/ 2 I*\/ 2 \/ 2 I*\/ 2
-1 + 1 - I + I + - ----- - ------- + - ----- + ------- + ----- - ------- + ----- + -------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(\left(-1 + 1\right) - i\right) + i\right)\right) + \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right)\right) + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)$$
=
0
$$0$$
product
/ ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\
| \/ 2 I*\/ 2 | | \/ 2 I*\/ 2 | |\/ 2 I*\/ 2 | |\/ 2 I*\/ 2 |
-(-I)*I*|- ----- - -------|*|- ----- + -------|*|----- - -------|*|----- + -------|
\ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \left(-1\right) i\right) \left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)$$
=
-1
$$-1$$
-1
Rapid solution
[src]
x1 = -1
$$x_{1} = -1$$
x2 = 1
$$x_{2} = 1$$
x3 = -I
$$x_{3} = - i$$
x4 = I
$$x_{4} = i$$
___ ___
\/ 2 I*\/ 2
x5 = - ----- - -------
2 2
$$x_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x6 = - ----- + -------
2 2
$$x_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x7 = ----- - -------
2 2
$$x_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x8 = ----- + -------
2 2
$$x_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
x8 = sqrt(2)/2 + sqrt(2)*i/2
Numerical answer
[src]
x1 = -0.707106781186548 - 0.707106781186548*i
x2 = 0.707106781186548 + 0.707106781186548*i
x3 = -0.707106781186548 + 0.707106781186548*i
x4 = 1.0*i
x5 = -1.0
x6 = 0.707106781186548 - 0.707106781186548*i
x7 = 1.0
x8 = -1.0*i
x8 = -1.0*i
The graph