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

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

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29 Dec 2013, 06:29
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If $$a^6-b^6=0$$ what is the value of $$a^3+b^3$$?

(1) $$a^3-b^3=-2$$
(2) $$ab<0$$
[Reveal] Spoiler: OA

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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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29 Dec 2013, 06:29
let's rephrase $$a^6-b^6=0$$ is equal to $$(a^3-b^3)(a^3+b^3)=0$$ and thus either $$a^3=b^3$$ or $$a^3=-b^3$$ or both.

1. $$a^3-b^3=-2$$ thus for $$(a^3-b^3)(a^3+b^3)=0$$ to be zero $$(a^3+b^3)=0$$
Sufficient.

2. ab<0 a and b have opposite sign thus $$a^3-b^3=0$$
Sufficient.

D
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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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29 Dec 2013, 13:19
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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 ?

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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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30 Dec 2013, 00:47
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mrwells2 wrote:
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 ?

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

$$a^6-b^6=0$$ --> apply $$x^2-y^2=(x-y)(x+y)$$: $$(a^3-b^3)(a^3+b^3)=0$$ --> the product of two multiples is 0, which implies that at least one of them must be 0: $$a^3-b^3=0$$ ($$a^3=b^3$$ --> $$a=b$$) or $$a^3+b^3=0$$.

(1) $$a^3-b^3=-2$$. Since $$a^3-b^3\neq{0}$$, then $$a^3+b^3=0$$. Sufficient.

(2) $$ab<0$$. This implies that a and b have opposite signs, thus $$a\neq{b}$$ ($$a^3-b^3\neq{0}$$). Therefore $$a^3+b^3=0$$. Sufficient.

Hope it's clear.
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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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30 Dec 2013, 00:55
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mrwells2 wrote:
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 ?

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.

mrwells2 wrote:
Does Statement #2 verify the assumption that a^3 = -b^3 ?

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.

mrwells2 wrote:
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 ?

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.
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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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31 Dec 2013, 09:00
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.

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Re: If a^6-b^6=0 what is the value of a^3+b^3? [#permalink]

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21 Nov 2017, 12:49
gmat6nplus1 wrote:
let's rephrase $$a^6-b^6=0$$ is equal to $$(a^3-b^3)(a^3+b^3)=0$$ and thus either $$a^3=b^3$$ or $$a^3=-b^3$$ or both.

1. $$a^3-b^3=-2$$ thus for $$(a^3-b^3)(a^3+b^3)=0$$ to be zero $$(a^3+b^3)=0$$
Sufficient.

2. ab<0 a and b have opposite sign thus $$a^3-b^3=0$$
Sufficient.

D

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

Cheers,
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Re: If a^6-b^6=0 what is the value of a^3+b^3?   [#permalink] 21 Nov 2017, 12:49
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