Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 3x^2=0
-
Equation 49x^3+14x^2+x=0
- Equation x^4-50x^2+49=0
-
Equation x*x*x=216
- Express {x} in terms of y where:
- 2*x+13*y=19
- 14*x-16*y=14
- 15*x+15*y=16
- 9*x-8*y=6
- Derivative of:
-
x*x*x
- Integral of d{x}:
-
x*x*x
- Canonical form:
- 216
-
Identical expressions
- x*x*x= two hundred and sixteen
- x multiply by x multiply by x equally 216
- x multiply by x multiply by x equally two hundred and sixteen
- xxx=216
-
Similar expressions
- x^x^x
- xxx-x-10=0
- xxx+12*6-24-9*4+72=0
- 2*16^(x-1/-4)-6/5*12^x+9^(x+1)=0
x*x*x=216 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x x x = 216$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{\left(1 x + 0\right)^{3}} = \sqrt[3]{216}$$
or
$$x = 6$$
We get the answer: x = 6
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{3} = 216$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = 216$$
where
$$r = 6$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = 1$$
so
$$\cos{\left(3 p \right)} = 1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3}$$
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:
$$z_{1} = 6$$
$$z_{2} = -3 - 3 \sqrt{3} i$$
$$z_{3} = -3 + 3 \sqrt{3} i$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = 6$$
$$x_{2} = -3 - 3 \sqrt{3} i$$
$$x_{3} = -3 + 3 \sqrt{3} i$$
$$x x x = 216$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{\left(1 x + 0\right)^{3}} = \sqrt[3]{216}$$
or
$$x = 6$$
We get the answer: x = 6
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{3} = 216$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = 216$$
where
$$r = 6$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = 1$$
so
$$\cos{\left(3 p \right)} = 1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3}$$
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:
$$z_{1} = 6$$
$$z_{2} = -3 - 3 \sqrt{3} i$$
$$z_{3} = -3 + 3 \sqrt{3} i$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = 6$$
$$x_{2} = -3 - 3 \sqrt{3} i$$
$$x_{3} = -3 + 3 \sqrt{3} i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -216$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -216$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -216$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -216$$
Rapid solution
[src]
x_1 = 6
$$x_{1} = 6$$
___ x_2 = -3 - 3*I*\/ 3
$$x_{2} = -3 - 3 \sqrt{3} i$$
___ x_3 = -3 + 3*I*\/ 3
$$x_{3} = -3 + 3 \sqrt{3} i$$
Sum and product of roots
[src]
sum
___ ___ 6 + -3 - 3*I*\/ 3 + -3 + 3*I*\/ 3
$$\left(6\right) + \left(-3 - 3 \sqrt{3} i\right) + \left(-3 + 3 \sqrt{3} i\right)$$
=
0
$$0$$
product
___ ___ 6 * -3 - 3*I*\/ 3 * -3 + 3*I*\/ 3
$$\left(6\right) * \left(-3 - 3 \sqrt{3} i\right) * \left(-3 + 3 \sqrt{3} i\right)$$
=
216
$$216$$
Numerical answer
[src]
x1 = -3.0 - 5.19615242270663*i
x2 = -3.0 + 5.19615242270663*i
x3 = 6.0
x3 = 6.0
The graph