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29 Dec 2013, 07:29
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D
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Difficulty:
95%
(hard)
Question Stats:
41% (01:56) correct
59%
(01:42)
wrong
based on 219
sessions
History
Date
Time
Result
Not Attempted Yet
If \(a^6-b^6=0\) what is the value of \(a^3+b^3\)?
(1) \(a^3-b^3=-2\)
(2) \(ab<0\)
(1) \(a^3-b^3=-2\)
(2) \(ab<0\)
30 Dec 2013, 01:47
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mrwells2
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.
Answer: D.
Hope it's clear.
General Discussion
29 Dec 2013, 07:29
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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
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
