Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2+2x+1=0
-
Equation x*7=56
-
Equation x^2=-5
-
Equation x*x=x
- Express {x} in terms of y where:
- -4*x+18*y=-2
- -2*x+2*y=-7
- -16*x+2*y=20
- -10*x-15*y=-2
-
Identical expressions
- x^ seven / fifteen = fifteen / twenty-eight
- x to the power of 7 divide by 15 equally 15 divide by 28
- x to the power of seven divide by fifteen equally fifteen divide by twenty minus eight
- x7/15=15/28
- x⁷/15=15/28
- x^7 divide by 15=15 divide by 28
x^7/15=15/28 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\frac{x^{7}}{15} = \frac{15}{28}$$
Because equation degree is equal to = 7 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 7-th degree of the equation sides:
We get:
$$\frac{\sqrt[7]{x^{7}}}{\sqrt[7]{15}} = \sqrt[7]{\frac{15}{28}}$$
or
$$\frac{15^{\frac{6}{7}} x}{15} = \frac{\sqrt[7]{56471520}}{14}$$
Expand brackets in the left part
Expand brackets in the right part
Divide both parts of the equation by 15^(6/7)/15
We get the answer: x = 847072800^(1/7)/14
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{7} = \frac{225}{28}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{7} e^{7 i p} = \frac{225}{28}$$
where
$$r = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
- the magnitude of the complex number
Substitute r:
$$e^{7 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(7 p \right)} + \cos{\left(7 p \right)} = 1$$
so
$$\cos{\left(7 p \right)} = 1$$
and
$$\sin{\left(7 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{7}$$
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:
$$z_{1} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
$$z_{2} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$z_{3} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$z_{4} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$z_{5} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$z_{6} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
$$z_{7} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
$$x_{2} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$x_{3} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$x_{4} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$x_{5} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$x_{6} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
$$x_{7} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
$$\frac{x^{7}}{15} = \frac{15}{28}$$
Because equation degree is equal to = 7 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 7-th degree of the equation sides:
We get:
$$\frac{\sqrt[7]{x^{7}}}{\sqrt[7]{15}} = \sqrt[7]{\frac{15}{28}}$$
or
$$\frac{15^{\frac{6}{7}} x}{15} = \frac{\sqrt[7]{56471520}}{14}$$
Expand brackets in the left part
x*15^6/7/15 = 56471520^(1/7)/14
Expand brackets in the right part
x*15^6/7/15 = 56471520^1/7/14
Divide both parts of the equation by 15^(6/7)/15
x = 56471520^(1/7)/14 / (15^(6/7)/15)
We get the answer: x = 847072800^(1/7)/14
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{7} = \frac{225}{28}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{7} e^{7 i p} = \frac{225}{28}$$
where
$$r = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
- the magnitude of the complex number
Substitute r:
$$e^{7 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(7 p \right)} + \cos{\left(7 p \right)} = 1$$
so
$$\cos{\left(7 p \right)} = 1$$
and
$$\sin{\left(7 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{7}$$
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:
$$z_{1} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
$$z_{2} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$z_{3} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$z_{4} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$z_{5} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$z_{6} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
$$z_{7} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
$$x_{2} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$x_{3} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
$$x_{4} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$x_{5} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
$$x_{6} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
$$x_{7} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
Rapid solution
[src]
2/7 7 _________
15 *\/ 3764768
x1 = -----------------
14
$$x_{1} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14}$$
2/7 7 _________ /pi\ 2/7 7 _________ /pi\
15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--|
\7 / \7 /
x2 = - ------------------------- - ---------------------------
14 14
$$x_{2} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
2/7 7 _________ /pi\ 2/7 7 _________ /pi\
15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--|
\7 / \7 /
x3 = - ------------------------- + ---------------------------
14 14
$$x_{3} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}$$
2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\
15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|
\ 7 / \ 7 /
x4 = --------------------------- - -----------------------------
14 14
$$x_{4} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\
15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|
\ 7 / \ 7 /
x5 = --------------------------- + -----------------------------
14 14
$$x_{5} = \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}$$
2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\
15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|
\ 7 / \ 7 /
x6 = - --------------------------- - -----------------------------
14 14
$$x_{6} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\
15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|
\ 7 / \ 7 /
x7 = - --------------------------- + -----------------------------
14 14
$$x_{7} = - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}$$
x7 = -15^(2/7)*3764768^(1/7)*cos(3*pi/7)/14 + 15^(2/7)*3764768^(1/7)*i*sin(3*pi/7)/14
Sum and product of roots
[src]
sum
2/7 7 _________ /pi\ 2/7 7 _________ /pi\ 2/7 7 _________ /pi\ 2/7 7 _________ /pi\ 2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\ 2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\
2/7 7 _________ 15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--| 15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--| 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----| 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----| 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----| 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|
15 *\/ 3764768 \7 / \7 / \7 / \7 / \ 7 / \ 7 / \ 7 / \ 7 / \ 7 / \ 7 / \ 7 / \ 7 /
----------------- + - ------------------------- - --------------------------- + - ------------------------- + --------------------------- + --------------------------- - ----------------------------- + --------------------------- + ----------------------------- + - --------------------------- - ----------------------------- + - --------------------------- + -----------------------------
14 14 14 14 14 14 14 14 14 14 14 14 14
$$\left(\left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}\right) + \left(\left(\left(\frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}\right) + \left(\left(\frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14} + \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}\right)\right) + \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}\right)\right)\right) + \left(\frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}\right)\right)\right) + \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}\right)$$
=
2/7 7 _________ /pi\ 2/7 7 _________ /3*pi\ 2/7 7 _________ /2*pi\
2/7 7 _________ 15 *\/ 3764768 *cos|--| 15 *\/ 3764768 *cos|----| 15 *\/ 3764768 *cos|----|
15 *\/ 3764768 \7 / \ 7 / \ 7 /
----------------- - ------------------------- - --------------------------- + ---------------------------
14 7 7 7
$$- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{7} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{7} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{7}$$
product
/ 2/7 7 _________ /pi\ 2/7 7 _________ /pi\\ / 2/7 7 _________ /pi\ 2/7 7 _________ /pi\\ / 2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\\ / 2/7 7 _________ /2*pi\ 2/7 7 _________ /2*pi\\ / 2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\\ / 2/7 7 _________ /3*pi\ 2/7 7 _________ /3*pi\\
2/7 7 _________ | 15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--|| | 15 *\/ 3764768 *cos|--| I*15 *\/ 3764768 *sin|--|| |15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|| |15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|| | 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----|| | 15 *\/ 3764768 *cos|----| I*15 *\/ 3764768 *sin|----||
15 *\/ 3764768 | \7 / \7 /| | \7 / \7 /| | \ 7 / \ 7 /| | \ 7 / \ 7 /| | \ 7 / \ 7 /| | \ 7 / \ 7 /|
-----------------*|- ------------------------- - ---------------------------|*|- ------------------------- + ---------------------------|*|--------------------------- - -----------------------------|*|--------------------------- + -----------------------------|*|- --------------------------- - -----------------------------|*|- --------------------------- + -----------------------------|
14 \ 14 14 / \ 14 14 / \ 14 14 / \ 14 14 / \ 14 14 / \ 14 14 /
$$\frac{15^{\frac{2}{7}} \sqrt[7]{3764768}}{14} \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}\right) \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{\pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{\pi}{7} \right)}}{14}\right) \left(\frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}\right) \left(\frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{2 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{2 \pi}{7} \right)}}{14}\right) \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} - \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}\right) \left(- \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} \cos{\left(\frac{3 \pi}{7} \right)}}{14} + \frac{15^{\frac{2}{7}} \sqrt[7]{3764768} i \sin{\left(\frac{3 \pi}{7} \right)}}{14}\right)$$
=
225 --- 28
$$\frac{225}{28}$$
225/28
Numerical answer
[src]
x1 = 0.839689198725172 - 1.05293695116514*i
x2 = 1.34675690961089
x3 = -0.299681605338734 - 1.31299090210339*i
x4 = -1.21338604819188 - 0.584335923624378*i
x5 = 0.839689198725172 + 1.05293695116514*i
x6 = -1.21338604819188 + 0.584335923624378*i
x7 = -0.299681605338734 + 1.31299090210339*i
x7 = -0.299681605338734 + 1.31299090210339*i