Can someone find the solutions and the magnitudes of the following problems(This is from my complex numbers unit):

1. (x^4)-1=0

2. (x^3)+1=0

3.(x^3)-1=0

4.(x^6)-1=0

5.(x^6)+1=0

Thank You!

Guest Feb 24, 2017

#1**0 **

1) -

Solve for x:

x^4 - 1 = 0

Add 1 to both sides:

x^4 = 1

Taking 4^th roots gives 1 times the 4^th roots of unity:

**Answer: |x = -1 or x = -i or x = i or x = 1**

**2) -**

Solve for x:

x^3 + 1 = 0

Subtract 1 from both sides:

x^3 = -1

Taking cube roots gives (-1)^(1/3) times the third roots of unity:

**Answer: |x = -1 or x = (-1)^(1/3) or x = -(-1)^(2/3)**

**3) -**

Solve for x:

x^3 + 1 = 0

Subtract 1 from both sides:

x^3 = -1

Taking cube roots gives (-1)^(1/3) times the third roots of unity:

**Answer: |x = -1 or x = (-1)^(1/3) or x = -(-1)^(2/3)**

**4) -**

Solve for x:

x^6 - 1 = 0

Add 1 to both sides:

x^6 = 1

Taking 6^th roots gives 1 times the 6^th roots of unity:

**Answer: |x = -1 or x = 1 or x = -(-1)^(1/3) or x = (-1)^(1/3) or x = -(-1)^(2/3) or x = (-1)^(2/3)**

**5) -**

Solve for x:

x^6 + 1 = 0

Subtract 1 from both sides:

x^6 = -1

Taking 6^th roots gives (-1)^(1/6) times the 6^th roots of unity:

**Answer: |x = -i or x = i or x = -(-1)^(1/6) or x = (-1)^(1/6) or x = -(-1)^(5/6) or x = (-1)^(5/6)**

Guest Feb 24, 2017