4th root of 1?

Answer 1

x= 1^1/4;
or x= 1^1/2*1^1/2;
or X= +-1*+-1
or X=1,-1.iota ,-iota

Answer 2

1

Answer 3

1

Answer 4

4th root of 1 is 1

Answer 5

Let x be the fourth root of unity , then by definition , we have
x^4 = 1
=> x^4 -1 = 0
=> (x^2 -1)( x^2 +1) = 0
=> either x^2 -1= 0 or x^2 +1 =0
=> either x= + 1 or -1 or x = +i or -i
Hence , fourth roots of unity are 1 , -1 , i , -i .

Answer 6

Recall that an "nth root of unity" is just another name for an nth root of one. The fourth roots are ±1, ±i, as noted earlier in the section on absolute value. We also saw that the eight 8th roots of unity when we looked at multiplication were ±1, ±i, and ±√2/2 ± i√2/2.

Let's consider now the sixth roots of unity. They will be placed around the circle at 60° intervals. Two of them, of course, are ±1. Let w be the one with argument 60°. The triangle with vertices at 0, 1, and w is an equilateral triangle, so it is easy to determine the coordinates of w. The x-coordinate is 1/2, and the y-coordinate is √3/2. Therefore, w is (1 + i√3)/2. The remaining sixth roots are reflections of w in the real and imaginary axes. In summary, the six sixth roots of unity are ±1, and (±1 ± i√3)/2 (where + and – can be taken in any order).

Now some of these sixth roots are lower roots of unity as well. The number –1 is a square root of unity, (–1 ± i√3)/2 are cube roots of unity, and 1 itself counts as a cube root, a square root, and a "first" root (anything is a first root of itself). But the remaining two sixth roots, namely, (1 ± i√3)/2, are sixth roots, but not any lower roots of unity. Such roots are called primitive, so (1 ± i√3)/2 are the two primitive sixth roots of unity.

It's fun to find roots of unity, but we've found most of the easy ones already.

Answer 7

1,-1,i,-i.

Answer 8

4th root of 1

=1
_
4_/ 1 =1
=>1^(1/4)=1
Multiply by 4 to the power of both sides.
=>1^(1/4)*(1/4)=1^4
=>1^1=1^4
=>1=1
Hence,4th root of 1 is 1.

*Use calculator:
press 1
tick square box before inverse.
press x^y
press 1/4(=0.25)
press =
you will get 1.

Answer 9

1=e^(2 pi i)
So the 4 roots are the exponentials where 2pi has been multiplied by 4/4, 3/4, 2/4, 1/4.

Answer 10

either 1 or -1