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I think it is B, but I also think that we are discussing semantics here.

Plugging in:
x^2= 1/2 , x = 1/ sqrt (2) , x^3= sqrt(2) / 4 This is not a fraction.
x^2= 9/4 , x = 3/ 2 , x^3= 27/8. This is a fraction.

Sqrt(2) / 4 is not a fraction because is not a rational number. Sqrt (2) is irrational, not an integer.

Rational numbers are those that can be written as a fraction of two integers (it will not be defined if the denominator =0) and rational numbers can be integers or fractions.

If x=n^(1/3), I think that x^2 can only be a fraction for those cases where n is a fraction where numerator and denominator are perfect cubes, and for these cases, x^3 will always be a fraction. All other values of n will give x^2 either integers or irrationals.

So could be D, depending on semantics. If we call sqrt (2) / 4 a fraction, then answer is B. But if we say that sqrt (2) / 4 is an irrational number and therefore not a fraction, then answer is B. So we just need to find out whatâ€™s ETS definition of fractional and irrational.

I'm a newcomer here.
Could anyone please clarify if there were any varants of answers in this subject.
Because I've not seen A) B) C) D) there and can't understand if sonmeone says "I think it's D" for example.

I think it is B, but I also think that we are discussing semantics here.

Plugging in: x^2= 1/2 , x = 1/ sqrt (2) , x^3= sqrt(2) / 4 This is not a fraction. x^2= 9/4 , x = 3/ 2 , x^3= 27/8. This is a fraction.

Sqrt(2) / 4 is not a fraction because is not a rational number. Sqrt (2) is irrational, not an integer.

Rational numbers are those that can be written as a fraction of two integers (it will not be defined if the denominator =0) and rational numbers can be integers or fractions.

If x=n^(1/3), I think that x^2 can only be a fraction for those cases where n is a fraction where numerator and denominator are perfect cubes, and for these cases, x^3 will always be a fraction. All other values of n will give x^2 either integers or irrationals.

So could be D, depending on semantics. If we call sqrt (2) / 4 a fraction, then answer is B. But if we say that sqrt (2) / 4 is an irrational number and therefore not a fraction, then answer is B. So we just need to find out whatâ€™s ETS definition of fractional and irrational.

Thank you Artabro for adding your opinion. Maybe we are over reading the question but this kind of point deserves to be understood.

I know that not every number can be described as a ratio of integers ; I do think sqrt can not be described in this manner as well. I was wondering if every number can result from a ratio of 2 relative numbers ?

Anyway, a "Fraction" is a ratio between two whole numbers.
Roots, like sqrt(2) or 2^(1/3) are Irrational numbers.
They can not be described as a ratio between two numbers.

So the answer is D.
It is true that we are talking semantics here, but ETS assumes that we are familiar with those definitions.

I guess we all learned something from this question

I just checked what ETS says about the numbers it uses.
It says "All numbers used are real".
So that'd mean they can be rational or irrational (square root 2, sqrt 3 etc).
So that means the ans to the original question would be B and not D.

Does everyone agree?
I think this is an important discussion because we need to be clear about this issue, esp. in DS where we can not assume ANYTHING!