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I disagree with explanations given above. Here is my approach:

Stmt (1) indicates that x=y otherwise x^6-y^6=0 does not hold true. Hence, x^3-y^3 is always equal to zero. SUFF.

Stmt (2) is clearly INSUFF because there is no inf. about x.

If my approach is wrong, I would appreciate any other explanations.

Yes, your approach is wrong: \(x^6-y^6=0\) implies that either \(x=y\) or \(x=-y\), for example \(1^6-1^6=0\) and also \(1^6-(-1)^6=0\). To see this algebraically you could rewrite \(x^6-y^6=0\) as \((x^3-y^3)(x^3+y^3)=0\) --> either \(x^3=y^3\), or \(x^3=-y^3\) --> so either \(x=y\) or \(x=-y\). OR \(x^6-y^6=0\) --> \(x^6=y^6\) --> \(x^2=y^2\) --> \(x=y\) or \(x=-y\).

Now, if \(x=y\) then \(x^3-y^3=0\) for any values of \(x\) and \(y\) BUT of \(x=-y\) then \(x^3-y^3=2x^3\) and we need the value of \(x\) (or \(y\)) the get the single numerical value of \(2x^3\). So statement (1) is not sufficient.

(2) y=0 --> clearly insufficient.

(1)+(2) \(y=0\), so \(x=0\) too (as \(x^6-y^6=0\)) and \(x^3-y^3=0\). Sufficient.

This is not tough question, but you need to be carefull not to fall into trap of ODD POWER of numbers. you always need to be suspicious when you see one.
_________________

IF... X = 0 Y = 0 0^3 - 0^3 = 0 and the answer is 0.

IF... X = 1 Y = -1 1^3 - (-1)^3 = 2 and the answer is 2. Fact 1 is INSUFFICIENT

2) Y = 0

This tells us NOTHING about the value of X, so there's no way to answer the question. Fact 2 is INSUFFICIENT

Combined, we know... X^6 - Y^6 = 0 Y = 0

Substituting in the value of Y, we have... X^6 = 0, so X MUST be 0. By extension, the answer to the question is ALWAYS 0^3 - 0^3 = 0. Combined, SUFFICIENT.

1- xˆ6 - yˆ6 = 0 As we have an even number (6) it means that X and Y can be either negative or positive. For example. X=3 Y=-3 If we put this values on the equations, then its going to be valid (equal 0)

However, if we put the same numbers on x^3 - yˆ3 we are going to have: 27 - ( -27) = 54

Therefore number 1 is not valid.

2-> This information alone doesn't help anything, we don't know about the X.

If we use both of them together, we are going to know that the only value will be 0. Therefore C