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C
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Difficulty:
75%
(hard)
Question Stats:
24% (01:08) correct
76%
(01:09)
wrong
based on 62
sessions
History
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Is \(m^n\) a perfect square?
(1) m is a perfect square
(2) n is a perfect square
(1) m is a perfect square
(2) n is a perfect square
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barryseal
\({m^n}\,\,\mathop = \limits^? \,\,{K^2}\,\,\,,\,\,\,K\,\,{\mathop{\rm int}}\)
\(\left( 1 \right)\,\,m = {J^2},\,\,J\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,{\mathop{\rm int}} \,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {4,0.5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,n = {W^2},\,\,W\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,{m^n} = {\left( {{J^2}} \right)^{{W^2}}} = {\left( {{J^{{W^2}}}} \right)^2}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {{J^{{W^2}}} = K} \right)\,\,\,\)
The correct answer is (C).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
POST-MORTEM (important to the most careful students): \(0^0\) is an undefined expression in Math (and also in the GMAT, of course),
hence the case \(m=0\) is implicitly associated with \(n \neq 0\). (The question stem implicitly implies that \(m^n\) exists!)
29 Nov 2018, 17:38
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barryseal
Since there's no information to suggest that m and n are integers, the correct answer is not A.
Statement 1: m is a perfect square
Consider these conflicting cases that satisfy statement 1:
CASE A) m = 9 and n = 2. In this case, m^n = 9^2 = 81. So, the answer to the target question is "YES m^n IS a perfect square."
CASE B) m = 9 and n = 0.5. In this case, m^n = 9^0.5 = 3. So, the answer to the target question is "NO m^n is NOT a perfect square."
Cheers,
Brent
