Question

(1)^1/3=?

Answer 1

I know simple, but can't think right now

Answer 2

im pretty sure it would be 1!

Answer 3

I was thinking 1/3 but not quite sure

Answer 4

nope lol can you try 1 then let me know because i took (1)^1/3 and i got 1 after i typed it in my calculator

Answer 5

On my last try, My calculator computes .333333 aka 1/3. try parenthesis around the one and then make sure you are placing 1/3 as a power/exponent outside the parenthesis

Answer 6

ok hold on lol

Answer 7

ok nevermind i got 1/3. this time lol

Answer 8

ok just making sure, how r u at trig?

Answer 9

yupp and ummm... it depends what the problem is lol

Answer 10

depends on what class you are taking
if you are using real numbers then
\[1^{\frac{1}{3}}=\sqrt[3]{1}=1\]

Answer 11

if you are living in the world of complex numbers, then there are three "cube roots" of one

Answer 12

complex

Answer 13

i'll display the whole problem

Answer 14

on of them is 1
the other two are \(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\) and its conjugate \(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\)

Answer 15

Solve the equation. (List your answers counterclockwise about the origin starting at the positive real axis. Express θ in radians.)

z3 − 1 = 0

Answer 16

there are a couple ways to find the other two.
from algebra you can solve
\[x^3=1\]
\[x^3-1=0\]
\[(x-1)(x^2+x+1)=0\] and use the quadratic formula for \(x^2+x+1=0\)

Answer 17

or you can use trig

Answer 18

or you can just look with your eyes at the unit circle divided in two three parts with one of the parts at 1

Answer 19

Its trig, using unit circle pi/3, pi, 5pi/3

Answer 20

no it would not be those
it would be the ones on the picture above
the angles are \(\theta=0,\theta=\frac{2\pi}{3},\theta=\frac{4\pi}{3}\)

Answer 21

you can check it is right using the quadratic formula, but we can work through the trig slowly if you like

Answer 22

the radius would b? the equation will be in this format, r(cos \[r(\cos \theta + isin \theta) ____ z ^{3}-1=0 _____ z ^{3}=1\]

Answer 23

ok i got it

Answer 24

the radius is 1

Answer 25

because you are trying to solve \(z^3=1\)

Answer 26

\[\sqrt[3]{z}= 1^{\frac{ 1 }{ 3 }} _____ 1/3(\cos \pi/3+isin \pi/3)\]

Answer 27

lets go slow, i sense there is some confusion about what you are trying to solve and how you write a number in trig form

Answer 28

the \(\frac{1}{3}\) is in the exponent, it is not a coefficient

Answer 29

your job is to solve \(z^3=1\) in the complex plane, i.e. your job is to find the three cube roots of one

we can write 1 as a complex number in the form of
\(1=\cos(0)+i\sin(0)\) which is a rather silly thing to do but it has it uses
we know this because \(\cos(0)=1\) and \(\sin(0)=0\) so this just says \(1=1+0i\)

Answer 30

to find the cube root, divide the angle by 3 and take the cube root of \(r\) which is in this case its 1

but zero divided by 3 is just zero, and the cube root of one is just one so again you get
\[\cos(0)+i\sin(0)=1\] which you already knew, since it is clear that \(1^3=1\)

the real job it to find the other two cubed roots of one

Answer 31

if you go around the unit circle again, you go from \(0\) to \(2\pi\) and so you can write \[1=\cos(2\pi)+i\sin(2\pi)\]
now again divide the angle by 3 for the next cubed root of 1 and get
\[\cos(\frac{2\pi}{3})+i\sin(\frac{2\pi}{3})\]

Answer 32

when you evaluate the trig functions you get
\[\cos(\frac{2\pi}{3})+i\sin(\frac{2\pi}{3})=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\]

Answer 33

let me know when i have lost you

Answer 34

got u its becoming clearer

Answer 35

Thank u so much

Answer 36

then if you go around the unit circle one more time, you get
\[1=\cos(4\pi)+i\sin(4\pi)\] divide by 3 get
\[\cos(\frac{4\pi}{3})+i\sin(\frac{4\pi}{3})\]

Answer 37

it is really much easier to visualize dividing the unit circle in to three parts
see the picture above