Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x-5=7
-
Equation 2*cos(x)+1=0
-
Equation x^2+8x+7=0
-
Equation x^6=5
- Express {x} in terms of y where:
- -4*x+20*y=10
- -13*x+7*y=11
- 14*x+16*y=14
- -16*x-7*y=-5
- Graphing y =:
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x^6
- Integral of d{x}:
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x^6
- Derivative of:
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x^6
-
Identical expressions
- x^ six = five
- x to the power of 6 equally 5
- x to the power of six equally five
- x6=5
- x⁶=5
-
Similar expressions
- x^6=64
- x^6=(6x-8)^3
- 80*x^6-80*x^4+16*x^2=0
x^6=5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{6} = 5$$
Because equation degree is equal to = 6 - contains the even number 6 in the numerator, then
the equation has two real roots.
Get the root 6-th degree of the equation sides:
We get:
$$\sqrt[6]{\left(1 x + 0\right)^{6}} = \sqrt[6]{5}$$
$$\sqrt[6]{\left(1 x + 0\right)^{6}} = - \sqrt[6]{5}$$
or
$$x = \sqrt[6]{5}$$
$$x = - \sqrt[6]{5}$$
Expand brackets in the right part
We get the answer: x = 5^(1/6)
Expand brackets in the right part
We get the answer: x = -5^(1/6)
or
$$x_{1} = - \sqrt[6]{5}$$
$$x_{2} = \sqrt[6]{5}$$
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = 5$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = 5$$
where
$$r = \sqrt[6]{5}$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = 1$$
so
$$\cos{\left(6 p \right)} = 1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3}$$
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:
$$z_{1} = - \sqrt[6]{5}$$
$$z_{2} = \sqrt[6]{5}$$
$$z_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt[6]{5}$$
$$x_{2} = \sqrt[6]{5}$$
$$x_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x^{6} = 5$$
Because equation degree is equal to = 6 - contains the even number 6 in the numerator, then
the equation has two real roots.
Get the root 6-th degree of the equation sides:
We get:
$$\sqrt[6]{\left(1 x + 0\right)^{6}} = \sqrt[6]{5}$$
$$\sqrt[6]{\left(1 x + 0\right)^{6}} = - \sqrt[6]{5}$$
or
$$x = \sqrt[6]{5}$$
$$x = - \sqrt[6]{5}$$
Expand brackets in the right part
x = 5^1/6
We get the answer: x = 5^(1/6)
Expand brackets in the right part
x = -5^1/6
We get the answer: x = -5^(1/6)
or
$$x_{1} = - \sqrt[6]{5}$$
$$x_{2} = \sqrt[6]{5}$$
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = 5$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = 5$$
where
$$r = \sqrt[6]{5}$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = 1$$
so
$$\cos{\left(6 p \right)} = 1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3}$$
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:
$$z_{1} = - \sqrt[6]{5}$$
$$z_{2} = \sqrt[6]{5}$$
$$z_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$z_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt[6]{5}$$
$$x_{2} = \sqrt[6]{5}$$
$$x_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
$$x_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
Sum and product of roots
[src]
sum
6 ___ ___ 6 ___ 6 ___ ___ 6 ___ 6 ___ ___ 6 ___ 6 ___ ___ 6 ___
6 ___ 6 ___ \/ 5 I*\/ 3 *\/ 5 \/ 5 I*\/ 3 *\/ 5 \/ 5 I*\/ 3 *\/ 5 \/ 5 I*\/ 3 *\/ 5
0 - \/ 5 + \/ 5 + - ----- - ------------- + - ----- + ------------- + ----- - ------------- + ----- + -------------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) + \left(\left(\left(\left(- \sqrt[6]{5} + 0\right) + \sqrt[6]{5}\right) - \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)\right) - \left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)\right)\right) + \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)$$
=
0
$$0$$
product
/ 6 ___ ___ 6 ___\ / 6 ___ ___ 6 ___\ /6 ___ ___ 6 ___\ /6 ___ ___ 6 ___\
6 ___ 6 ___ | \/ 5 I*\/ 3 *\/ 5 | | \/ 5 I*\/ 3 *\/ 5 | |\/ 5 I*\/ 3 *\/ 5 | |\/ 5 I*\/ 3 *\/ 5 |
1*-\/ 5 *\/ 5 *|- ----- - -------------|*|- ----- + -------------|*|----- - -------------|*|----- + -------------|
\ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\sqrt[6]{5} \cdot 1 \left(- \sqrt[6]{5}\right) \left(- \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(- \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)$$
=
-5
$$-5$$
-5
Rapid solution
[src]
6 ___ x1 = -\/ 5
$$x_{1} = - \sqrt[6]{5}$$
6 ___ x2 = \/ 5
$$x_{2} = \sqrt[6]{5}$$
6 ___ ___ 6 ___
\/ 5 I*\/ 3 *\/ 5
x3 = - ----- - -------------
2 2
$$x_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
6 ___ ___ 6 ___
\/ 5 I*\/ 3 *\/ 5
x4 = - ----- + -------------
2 2
$$x_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
6 ___ ___ 6 ___
\/ 5 I*\/ 3 *\/ 5
x5 = ----- - -------------
2 2
$$x_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
6 ___ ___ 6 ___
\/ 5 I*\/ 3 *\/ 5
x6 = ----- + -------------
2 2
$$x_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}$$
Numerical answer
[src]
x1 = -0.653830243005915 - 1.13246720041135*i
x2 = 0.653830243005915 + 1.13246720041135*i
x3 = -1.30766048601183
x4 = 1.30766048601183
x5 = 0.653830243005915 - 1.13246720041135*i
x6 = -0.653830243005915 + 1.13246720041135*i
x6 = -0.653830243005915 + 1.13246720041135*i
The graph