If a^6-b^6=0 what is the value of a^3+b^3?

When reverse factoring a^6 - b^6 to (a^3 - b^3) (a^3 + b^3) how do you come to the conclusion that a^3 must equal either b^3 or -b^3 ? Is this a rule that can be applied when factoring any equation? Or just applicable because the equation is = 0 ?

Does Statement #2 verify the assumption that a^3 = -b^3 ?

Additionally, how can you be certain on statement 2 alone? Is it because (a^3 - b^3) = positive number assuming a = -b^3 and a is positive. then (a^3 + b^3) = zero, assuming a = -b^3 and a is positive ?

[EDIT]

Based on statement two you can infer that a is positive and that b is negative.

Therefore (a^3 -(-b)^3) = positive number or 2(a)^3.

And that (a^3 + (-b)^3) = 0 ?

the equation equals to zero whenever that happen... it's like having (x-3)(x+2)=0 the equation yields zero when x=3 [(3-3)(3+2)=0] or when x=-2 (plug in) here we are doing the same thing but with letters. In particular whenever \(a^3=|b^3|\) then our equation will hold true because the result will be zero.



statement two says that a and b have opposit sign, refer back to our two solutions and check what happens whenever a and b have opposite sign (odd exponents don't hide the sign).
If \(a^3=-b^3 ------> a^3+b^3=0 ------> (a^3-b^3)(0)=0\) the equation is valid.



it doesn't necessarely have to be b the negative one. ab<0 case1: a=-ve b=+ve case2: a=+ve b=-ve (and viceversa)
ab<0 tells us that the variables have opposite sign and that they are different from zero.

Hope it shoves some haze away. Let me know.

If a^6-b^6=0 what is the value of a^3+b^3?

(1) a^3-b^3=-2
(2) ab<0

Given, \(a^6 - b^6 = 0\)

Statement I:

\(a^6 - b^6 = 0\)

Or, \((a ^ 3 - b^3) (a ^ 3 + b^3) = 0\)

As per statement (1), \(a^3-b^3=-2\) , hence \((a ^ 3 + b^3) = 0\) to make \(a^6 - b^6 = 0\)

Hence, statement (1) sufficient.

Statement 2:

\(a^6 - b^6 = 0\)

Or, \(a ^ 6 = b ^ 6\)

Or, \((a^3) (a ^3) = (b^3)(b^3)\) ....(A)

As per statement 2, \(ab<0\)
Or, a and b are of opposite sign
Or, a ^3 and b ^3 are of opposite sign ...(B)

From (A) & (B) above, \((a^3) = - (b^3)\)
Or, \(a^3 + b^3 = 0\)

Hence, statement (2) sufficient.

Answer : (D)

For the second statement, if ab<0 then a^3+b^3=0 not the other factor.

Cheers,
Mad :D

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