# Find the 5 roots of x^5+1=0 in polar and Cartesian form. (x^5 means x to the 5th power)

x^5 = -1 = e^[i (2n + 1) pi]
where i is any integer
x = [e^[i (2n + 1) pi]]^(1/5)
= e^(i pi/5)= cos pi/5 + i sin (pi/5)
= e^(3 i pi/5)
= cos (2 pi/5) + i sin (2 pi/5)
etc.

## To find the roots of the equation x^5 + 1 = 0 in polar and Cartesian form, we can follow these steps:

Step 1: Express -1 in polar form:
-1 can be written in polar form as e^(iπ), where i is the imaginary unit.

Step 2: Apply De Moivre's theorem:
Using De Moivre's theorem, we raise e^(iπ) to the power of (1/5). This gives us:
x = [e^(iπ)]^(1/5)

Step 3: Simplify the expression:
Applying the exponent rule, we get:
x = e^(iπ/5)

Step 4: Convert to Cartesian form:
Using Euler's formula, e^(ix) = cos x + i sin x, we can rewrite e^(iπ/5) as:
x = cos(π/5) + i sin(π/5)

Step 5: Find the other roots:
Since we are dealing with a fifth-degree equation, there will be a total of 5 roots. To obtain the other roots, we can use the fact that the solutions will be evenly distributed around the unit circle.

The other roots can be found by multiplying the first root, e^(iπ/5), by appropriate powers of e^(2πi/5):
x = cos(3π/5) + i sin(3π/5)
x = cos(5π/5) + i sin(5π/5) (which simplifies to -1)
x = cos(7π/5) + i sin(7π/5)
x = cos(9π/5) + i sin(9π/5)

So, the 5 roots of the equation x^5 + 1 = 0 are:
1. x = cos(π/5) + i sin(π/5)
2. x = cos(3π/5) + i sin(3π/5)
3. x = -1
4. x = cos(7π/5) + i sin(7π/5)
5. x = cos(9π/5) + i sin(9π/5)

These roots are expressed in both polar and Cartesian form.