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What is the value of a^3+b^3 ?

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What is the value of a^3+b^3 ?  [#permalink]

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New post 14 Feb 2019, 13:12
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Question Stats:

57% (01:41) correct 43% (01:29) wrong based on 28 sessions

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GMATH practice exercise (Quant Class 14)

What is the value of \({a^3} + {b^3}\) ?

\(\left( 1 \right)\,\,a + b = 1\)
\(\left( 2 \right)\,\,{a^2} + {b^2} = 2\)

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Re: What is the value of a^3+b^3 ?  [#permalink]

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New post 14 Feb 2019, 14:58
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fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of \({a^3} + {b^3}\) ?

\(\left( 1 \right)\,\,a + b = 1\)
\(\left( 2 \right)\,\,{a^2} + {b^2} = 2\)


Statement 1:
Case 1: a=0 and b=1, with the result that a+b = 0+1 = 1
In this case, a³+b³ = 0³ + 1³ = 1
Case 2: a=2 and b=-1, with the result that a+b = 2-1 = 1
In this case, a³+b³ = 2³ + (-1)³ = 7
Since a³+b³ can be different values, INSUFFICIENT.

Statement 2:
Case 3: a=1 and b=1, with the result that a²+b² = 1²+1² = 2
In this case, a³+b³ = 1³ + 1³ = 2
Case 4: a=-1 and b=-1, with the result that a²+b² = (-1)²+(-1)² = 2
In this case, a³+b³ = (-1)³ + (-1)³ = -2
Since a³+b³ can be different values, INSUFFICIENT.

Statements combined:
Squaring a+b = 1, we get:
a² + b² + 2ab = 1

Substituting a²+b²=2 into the red equation above, we get:
2 + 2ab = 1
2ab = -1
ab = -0.5

Multiplying a²+b²=2 by a+b=1, we get:
(a²+b²)(a+b) = 2*1
a³ + b³ + a²b + ab² = 2
a³ + b³ + ab(a+b) = 2

Substituting ab=-0.5 and a+b=1 into the blue equation above, we get:
a³ + b³ + (-0.5)(1) = 2
a³ + b³ = 2.5
SUFFICIENT.


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Re: What is the value of a^3+b^3 ?  [#permalink]

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New post 14 Feb 2019, 18:35
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of \({a^3} + {b^3}\) ?

\(\left( 1 \right)\,\,a + b = 1\)
\(\left( 2 \right)\,\,{a^2} + {b^2} = 2\)

\(? = {a^3} + {b^3}\)


\(\left( 1 \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,? = 1 \hfill \cr
\,{\rm{Take}}\,\left( {a,b} \right) = \left( {2, - 1} \right)\,\,\,\, \Rightarrow \,\,\,? \ne 1 \hfill \cr} \right.\)

\(\left( 2 \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 2 \hfill \cr
\,{\rm{Take}}\,\left( {a,b} \right) = \left( {\sqrt 2 ,0} \right)\,\,\,\, \Rightarrow \,\,\,? \ne 2 \hfill \cr} \right.\)


\(\left( {1 + 2} \right)\,\,\,2 = \left( {a + b} \right)\left( {{a^2} + {b^2}} \right) = \underbrace {{a^3} + {b^3}}_{{\rm{focus}}} + ab\underbrace {\left( {a + b} \right)}_{ = \,1}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{?_{{\rm{temporary}}}} = ab\)

\(\left\{ \matrix{
\,{1^2} = {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} \hfill \cr
\,2 = {a^2} + {b^2} \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,2ab = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{?_{{\rm{temporary}}}}\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)


The correct answer is (C).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

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Re: What is the value of a^3+b^3 ?   [#permalink] 14 Feb 2019, 18:35
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