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Equation:
Equation z^5+32=0
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Similar expressions
z^5-32=0

z^5+32=0 equation

The teacher will be very surprised to see your correct solution 😉

The solution

You have entered [src]

 5         
z  + 32 = 0

z^{5} + 32 = 0

Simplify

Detail solution

Given the equation

z^{5} + 32 = 0

Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:

\sqrt[5]{\left(1 z + 0\right)^{5}} = \sqrt[5]{-32}

z = 2 \sqrt[5]{-1}

Expand brackets in the right part

z = -2*1^1/5

We get the answer: z = 2*(-1)^(1/5)

All other 4 root(s) is the complex numbers.
do replacement:

w = z

then the equation will be the:

w^{5} = -32

Any complex number can presented so:

w = r e^{i p}

substitute to the equation

r^{5} e^{5 i p} = -32

where

r = 2

- the magnitude of the complex number
Substitute r:

e^{5 i p} = -1

Using Euler’s formula, we find roots for p

i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1

\cos{\left(5 p \right)} = -1

and

\sin{\left(5 p \right)} = 0

then

p = \frac{2 \pi N}{5} + \frac{\pi}{5}

where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for w
Consequently, the solution will be for w:

w_{1} = -2

w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}

w_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}

w_{4} = \frac{1}{2} + 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}

w_{5} = - 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}

do backward replacement

w = z

z = w

The final answer:

z_{1} = -2

z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}

z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}

z_{4} = \frac{1}{2} + 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}

z_{5} = - 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}

Rapid solution [src]

z1 = -2

z_{1} = -2

Simplify

                   /       ___________              ___________\
           ___     |      /       ___              /       ___ |
     1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  |
z2 = - - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- |
     2     2       \  \/    8     8            \/    8     8   /

z_{2} = - \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right)

Simplify

           /       ___________        ___________              ___________              ___________\                                      
           |      /       ___        /       ___              /       ___              /       ___ |                                      
           |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________
           |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ 
     1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  
z3 = - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + ----- 
     2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8

z_{3} = \frac{1}{2} + 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)

Simplify

           /     ___________        ___________              ___________              ___________\                                      
           |    /       ___        /       ___              /       ___              /       ___ |                                      
           |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________
           |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ 
     1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  
z4 = - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + ----- 
     2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8

z_{4} = - 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)

Simplify

                          ___________
           ___           /       ___ 
     1   \/ 5           /  5   \/ 5  
z5 = - + ----- + 2*I*  /   - - ----- 
     2     2         \/    8     8

z_{5} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}

Simplify

Sum and product of roots [src]

sum

                                                                            /       ___________        ___________              ___________              ___________\                                               /     ___________        ___________              ___________              ___________\                                                                         
                                                                            |      /       ___        /       ___              /       ___              /       ___ |                                               |    /       ___        /       ___              /       ___              /       ___ |                                                                         
                      /       ___________              ___________\         |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________         |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________                        ___________
              ___     |      /       ___              /       ___ |         |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___          |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___          ___           /       ___ 
        1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  |   1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5     1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5     1   \/ 5           /  5   \/ 5  
0 - 2 + - - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- | + - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + -----  + - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + -----  + - + ----- + 2*I*  /   - - ----- 
        2     2       \  \/    8     8            \/    8     8   /   2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8      2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8      2     2         \/    8     8

\left(\left(\left(\left(-2 + 0\right) + \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right)\right) + \left(\frac{1}{2} + 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right) + \left(- 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right)

                                                    /     ___________        ___________              ___________              ___________\     /       ___________        ___________              ___________              ___________\                       
                                                    |    /       ___        /       ___              /       ___              /       ___ |     |      /       ___        /       ___              /       ___              /       ___ |                       
  /       ___________              ___________\     |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |     |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |            ___________
  |      /       ___              /       ___ |     |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |     |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |           /       ___ 
  |     /  5   \/ 5       ___    /  5   \/ 5  |     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |          /  5   \/ 5  
I*|-   /   - - -----  - \/ 5 *  /   - - ----- | + I*|---------------- - ---------------- + ---------------------- + ----------------------| + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*I*  /   - - ----- 
  \  \/    8     8            \/    8     8   /     \       2                  2                     2                        2           /     \         2                  2                     2                        2           /       \/    8     8

i \left(- \sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right) + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)

product

                                                                   /      /       ___________        ___________              ___________              ___________\                                      \ /      /     ___________        ___________              ___________              ___________\                                      \                                   
                                                                   |      |      /       ___        /       ___              /       ___              /       ___ |                                      | |      |    /       ___        /       ___              /       ___              /       ___ |                                      |                                   
     /              /       ___________              ___________\\ |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________| |      |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________| /                     ___________\
     |      ___     |      /       ___              /       ___ || |      |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ | |      |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ | |      ___           /       ___ |
     |1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  || |1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  | |1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  | |1   \/ 5           /  5   \/ 5  |
1*-2*|- - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- ||*|- + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + ----- |*|- + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + ----- |*|- + ----- + 2*I*  /   - - ----- |
     \2     2       \  \/    8     8            \/    8     8   // \2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8   / \2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8   / \2     2         \/    8     8   /

1 \left(-2\right) \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(\frac{1}{2} + 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(- 2 \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right)

           _______________                                 ______________
          /           ___         _______________         /          ___ 
      I*\/  50 - 10*\/ 5         /           ___    5*I*\/  10 - 2*\/ 5  
-32 + -------------------- - I*\/  50 + 10*\/ 5   + ---------------------
               2                                              2

-32 - i \sqrt{10 \sqrt{5} + 50} + \frac{i \sqrt{- 10 \sqrt{5} + 50}}{2} + \frac{5 i \sqrt{- 2 \sqrt{5} + 10}}{2}

-32 + i*sqrt(50 - 10*sqrt(5))/2 - i*sqrt(50 + 10*sqrt(5)) + 5*i*sqrt(10 - 2*sqrt(5))/2

Numerical answer [src]

z1 = -2.0

z2 = -0.618033988749895 - 1.90211303259031*i

z3 = 1.61803398874989 + 1.17557050458495*i

z4 = -0.618033988749895 + 1.90211303259031*i

z5 = 1.61803398874989 - 1.17557050458495*i

z5 = 1.61803398874989 - 1.17557050458495*i

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Identical expressions

Similar expressions

z^5+32=0 equation

Numerical solution:

The solution