What are the five answers to this equations?

Hi Gibsonj338      

What are the five answers to this equations?  Please show how you got to your answers.

\(x^5=243\)

3 is the first obvious solution.

Now, there will be 5 roots altogether.

each of these will be 3 units from the origin of the real/imaginary plane and they will be spaced at 2pi/5 radians intervals

\(\mbox{The 5 roots in polar form are:}\\~\\ 3,\; 3e^{2\pi i/5},\;3e^{4\pi i/5},\;3e^{6\pi i/5},\;3e^{8\pi i/5}\\~\\ 3,\;\; 3\left[cos(\frac{2\pi}{5})+isin(\frac{2\pi}{5})\right] ,\;\; 3\left[cos(\frac{4\pi}{5})+isin(\frac{4\pi}{5})\right] ,\;\; 3\left[cos(\frac{6\pi}{5})+isin(\frac{6\pi}{5})\right] ,\;\; 3\left[cos(\frac{8\pi}{5})+isin(\frac{8\pi}{5})\right]\\~\\ 3,\;\; 3\left[cos(72^0)+isin(72^0)\right] ,\;\; 3\left[cos(144^0)+isin(144^0)\right] ,\;\; 3\left[cos(216^0)+isin(216^0)\right] ,\;\; 3\left[cos(288^0)+isin(288^0)\right]\\~\\ 3,\;\; 3\left[0.3090+0.9511i\right] ,\;\; 3\left[-0.8090+0.5878i\right] ,\;\; 3\left[-0.9090-0.5878i\right] ,\;\; 3\left[0.3090-0.9511i\right]\\~\\ \mbox{So the 5 roots are:}\\~\\ 3,\;\;\quad 0.3090+2.8532i ,\;\; \:\quad -2.4271+1.7634i ,\;\; \;\quad -2.4271-1.7634i ,\;\; \;\quad 0.9271-2.8532i\\~\\\)

These are the approximations  :)

The last guest answer gives the exact values.

More info and graph here.

http://www.wolframalpha.com/input/?i=x%5E5%3D243

P. S. It has ONLY 3 roots, since 3 the highest power of x. One "real" and two "complex". It is highly involved and was generated by computer.

Solve for x:
x^3+x^2 = 243

Subtract 243 from both sides:
x^3+x^2-243 = 0

Eliminate the quadratic term by substituting y = x+1/3:
-243+(y-1/3)^2+(y-1/3)^3 = 0

Expand out terms of the left hand side:
y^3-y/3-6559/27 = 0

If y = z+lambda/z then z = 1/2 (y+sqrt(y^2-4 lambda)) which will be used during back substitution:
-6559/27+1/3 (-z-lambda/z)+(z+lambda/z)^3 = 0

Multiply both sides by z^3 and collect in terms of z:
-(6559 z^3)/27+z^6+lambda^3+z^4 (3 lambda-1/3)+z^2 (3 lambda^2-lambda/3) = 0

Substitute lambda = 1/9 and then u = z^3, yielding a quadratic equation in the variable u:
u^2-(6559 u)/27+1/729 = 0

Find the positive solution to the quadratic equation:
u = 1/54 (6559+81 sqrt(6557))

Substitute back for u = z^3:
z^3 = 1/54 (6559+81 sqrt(6557))

Taking cube roots gives (6559+81 sqrt(6557))^(1/3)/(3 2^(1/3)) times the third roots of unity:
z = 1/3 (1/2 (6559+81 sqrt(6557)))^(1/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Substitute back for z = y/2+1/2 sqrt(y^2-(4)/9):
y/2+1/2 sqrt(y^2-(4)/9) = 1/3 (1/2 (6559+81 sqrt(6557)))^(1/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Rewrite the left hand side by combining fractions. y/2+1/2 sqrt(y^2-(4)/9)  =  1/6 (3 y+sqrt(9 y^2-4)):
1/6 (3 y+sqrt(9 y^2-4)) = 1/3 (1/2 (6559+81 sqrt(6557)))^(1/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Multiply both sides by 6:
3 y+sqrt(9 y^2-4) = 2^(2/3) (6559+81 sqrt(6557))^(1/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 3 y from both sides:
sqrt(9 y^2-4) = 2^(2/3) (6559+81 sqrt(6557))^(1/3)-3 y or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Raise both sides to the power of two:
9 y^2-4 = (2^(2/3) (6559+81 sqrt(6557))^(1/3)-3 y)^2 or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Expand out terms of the right hand side:
9 y^2-4 = 9 y^2-6 2^(2/3) (6559+81 sqrt(6557))^(1/3) y+2 2^(1/3) (6559+81 sqrt(6557))^(2/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 9 y^2-6 2^(2/3) (6559+81 sqrt(6557))^(1/3) y+2 2^(1/3) (6559+81 sqrt(6557))^(2/3) from both sides:
6 2^(2/3) (6559+81 sqrt(6557))^(1/3) y-4-2 2^(1/3) (6559+81 sqrt(6557))^(2/3) = 0 or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Factor constant terms from the left hand side:
2 (3 2^(2/3) (6559+81 sqrt(6557))^(1/3) y-2-2^(1/3) (6559+81 sqrt(6557))^(2/3)) = 0 or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Divide both sides by 2:
3 2^(2/3) (6559+81 sqrt(6557))^(1/3) y-2-2^(1/3) (6559+81 sqrt(6557))^(2/3) = 0 or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Add 2+2^(1/3) (6559+81 sqrt(6557))^(2/3) to both sides:
3 2^(2/3) (6559+81 sqrt(6557))^(1/3) y = 2+2^(1/3) (6559+81 sqrt(6557))^(2/3) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Divide both sides by 3 2^(2/3) (6559+81 sqrt(6557))^(1/3):
y = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Substitute back for y = x+1/3:
x+1/3 = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 1/3 from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or z = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Substitute back for z = y/2+1/2 sqrt(y^2-(4)/9):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or y/2+1/2 sqrt(y^2-(4)/9) = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Rewrite the left hand side by combining fractions. y/2+1/2 sqrt(y^2-(4)/9)  =  1/6 (3 y+sqrt(9 y^2-4)):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 1/6 (3 y+sqrt(9 y^2-4)) = -1/3 (1/2 (-6559-81 sqrt(6557)))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Multiply both sides by 6:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 3 y+sqrt(9 y^2-4) = -2^(2/3) (-6559-81 sqrt(6557))^(1/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 3 y from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or sqrt(9 y^2-4) = -2^(2/3) (-6559-81 sqrt(6557))^(1/3)-3 y or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Raise both sides to the power of two:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 9 y^2-4 = (-2^(2/3) (-6559-81 sqrt(6557))^(1/3)-3 y)^2 or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Expand out terms of the right hand side:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 9 y^2-4 = 9 y^2+6 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y+2 2^(1/3) (-6559-81 sqrt(6557))^(2/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 9 y^2+6 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y+2 2^(1/3) (-6559-81 sqrt(6557))^(2/3) from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or -6 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y-4-2 2^(1/3) (-6559-81 sqrt(6557))^(2/3) = 0 or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Factor constant terms from the left hand side:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or -2 (3 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y+2+2^(1/3) (-6559-81 sqrt(6557))^(2/3)) = 0 or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Divide both sides by -2:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 3 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y+2+2^(1/3) (-6559-81 sqrt(6557))^(2/3) = 0 or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 2+2^(1/3) (-6559-81 sqrt(6557))^(2/3) from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or 3 2^(2/3) (-6559-81 sqrt(6557))^(1/3) y = -2-2^(1/3) (-6559-81 sqrt(6557))^(2/3) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Divide both sides by 3 2^(2/3) (-6559-81 sqrt(6557))^(1/3):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or y = -((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Substitute back for y = x+1/3:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x+1/3 = -((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Subtract 1/3 from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or z = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Substitute back for z = y/2+1/2 sqrt(y^2-(4)/9):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or y/2+1/2 sqrt(y^2-(4)/9) = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Rewrite the left hand side by combining fractions. y/2+1/2 sqrt(y^2-(4)/9)  =  1/6 (3 y+sqrt(9 y^2-4)):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 1/6 (3 y+sqrt(9 y^2-4)) = 1/3 (-1)^(2/3) (1/2 (6559+81 sqrt(6557)))^(1/3)

Multiply both sides by 6:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 3 y+sqrt(9 y^2-4) = (-2)^(2/3) (6559+81 sqrt(6557))^(1/3)

Subtract 3 y from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or sqrt(9 y^2-4) = (-2)^(2/3) (6559+81 sqrt(6557))^(1/3)-3 y

Raise both sides to the power of two:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 9 y^2-4 = ((-2)^(2/3) (6559+81 sqrt(6557))^(1/3)-3 y)^2

Expand out terms of the right hand side:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 9 y^2-4 = 9 y^2-6 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y-2 (-2)^(1/3) (6559+81 sqrt(6557))^(2/3)

Subtract 9 y^2-6 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y-2 (-2)^(1/3) (6559+81 sqrt(6557))^(2/3) from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 6 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y-4+2 (-2)^(1/3) (6559+81 sqrt(6557))^(2/3) = 0

Factor constant terms from the left hand side:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 2 (3 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y-2+(-2)^(1/3) (6559+81 sqrt(6557))^(2/3)) = 0

Divide both sides by 2:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 3 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y-2+(-2)^(1/3) (6559+81 sqrt(6557))^(2/3) = 0

Subtract (-2)^(1/3) (6559+81 sqrt(6557))^(2/3)-2 from both sides:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or 3 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3) y = 2-(-2)^(1/3) (6559+81 sqrt(6557))^(2/3)

Divide both sides by 3 (-2)^(2/3) (6559+81 sqrt(6557))^(1/3):
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or y = 1/3 (-1)^(2/3) ((6559+81 sqrt(6557))/(2))^(1/3)-1/3 ((-2)/(6559+81 sqrt(6557)))^(1/3)

Substitute back for y = x+1/3:
x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3 or x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3)) or x+1/3 = 1/3 (-1)^(2/3) ((6559+81 sqrt(6557))/(2))^(1/3)-1/3 ((-2)/(6559+81 sqrt(6557)))^(1/3)

Subtract 1/3 from both sides:
Answer: |  x = (2+2^(1/3) (6559+81 sqrt(6557))^(2/3))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))-1/3    or    x = -1/3-((-1)^(2/3) (-2-2^(1/3) (-6559-81 sqrt(6557))^(2/3)))/(3 2^(2/3) (6559+81 sqrt(6557))^(1/3))    or    x = 1/3 (-1)^(2/3) ((6559+81 sqrt(6557))/(2))^(1/3)+(-1/3-1/3 (-2/(6559+81 sqrt(6557)))^(1/3))

Here are the 3 solutions in approximated forms in answer #1 above.

1) x=~  5.9241 (Real)

2) x=~ -3.4620-5.3882 i

3) x=~ -3.4620+5.3882 i

Here is the "Real" solution:

Solve for x over the real numbers:
x^5 = 243

Take 5^th roots of both sides:
Answer: |  x = 3

Here are the "complex solutions.

Solve for x:
x^5 = 243

Taking 5^th roots gives 3 times the 5^th roots of unity:
Answer: |  x = 3    or    x = -3 (-1)^(1/5)    or    x = 3 (-1)^(2/5)    or    x = -3 (-1)^(3/5)    or    x = 3 (-1)^(4/5)