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!
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)
