How many roots does this equation x^(2/3) = 1 have?

Question

A polynomial equation of nth degree has n roots. Does that mean this equation has 2/3rds roots? What are the roots, which may include complex roots?

Answer 1

Good question.

Let 1 = exp(2*i*pi + 2*k*i*pi)

raising this to the 3/2 power:

x = exp(3*i*pi + 3*k*i*pi)

which has values
1 and -1

So, I think that the original equation is equivalent to
x^2 = 1 which, of course, has 2 solutions.

I'm a little surprized that the solutions are not the three roots of 1. I'll have to give it some more thought.

Answer 2

it is correct that a polynomial equation of nth degree has n roots. but another condition for a polynomial is that the exponents of x should be whole numbers. hence x^2/3 = 1 is not a polynomial.

The root of the equation is 1. maybe some complex roots also

Answer 3

This equation does not have 2/3rd roots, certainly not. An equation can have either 0 or a positive integral number of roots. To solve the given equation, we have to re-write it as: x=1^(3/2). The only solution u get is x=1. There are no complex solutions.

Answer 4

Raise both sides of the equation to the power 3/2. This will eliminate the fractional power on the right side, and since 1 to any power is just 1...you will have your answer for x.

x = 1

Work:
x^(2/3) = 1
[x^(2/3)]^(3/2) = 1^(3/2)
x^(6/6) = 1
x^1 = 1
x = 1

Answer 5

only 2 roots
x^2=1^3
x^2=1
x1=-1
x2=1

The proposition "A polynomial equation of nth degree has n roots." is wrong
It must be "may have at most n roots"

For the second degree equations,
if diskriminant is negative
its roots will be complex roots.

Answer 6

You rewrite it as (x^1/3)^2

So, you know that x^1/3= +1 or -1
cube roots have three roots, and there are two in this, leaving a total of 6 roots for the original equation, some of which might be the same.

For fractional exponents, the number of roots will be n * m, where n and m are the numerator and denominator of the exponent in lowest terms.

Answer 7

the only answer is 1

x^(2/3) = 1
x = 2/3 root of 1
x = 1

Answer 8

Cube both sides to get x^2 = 1. Solve that. Remember to discard any and all extraneous solution (if there are any).