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16 Sep 2014, 01:24
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Difficulty:
55%
(hard)
Question Stats:
58% (01:42) correct
42%
(01:49)
wrong
based on 271
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If \(x=\sqrt[5]{-37}\) then which of the following must be true?
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
16 Sep 2014, 01:24
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Official Solution:
If \(x=\sqrt[5]{-37}\) then which of the following must be true?
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
MUST KNOW FOR THE GMAT:
• Even roots from a negative number are undefined on the GMAT (as GMAT is dealing only with Real Numbers): \(\sqrt[{even}]{negative}=undefined\), for example, \(\sqrt{-25}=undefined\).
• Odd roots have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).
BACK TO THE ORIGINAL QUESTION:
As \(-2^5=-32\), then \(x\) must be a little bit less than -2, hence \(x=\sqrt[5]{-37} \approx -2.1 \lt -2\). Thus \(x^3 \approx (-2.1)^3 \approx -8.something \lt -8\), so option D must be true.
As for the other options:
A. \(\sqrt{-x}=\sqrt{-(-2.1)}=\sqrt{2.1} \lt 2\), \(\sqrt{-x} \gt 2\) is not true.
B. \(x \approx -2.1 \lt -2\), thus \(x \gt -2\) is also not true.
C. \(x^2 \approx (-2.1)^2=4.something \gt 4\), thus \(x^2 \lt 4\) is also not true.
Answer: D
If \(x=\sqrt[5]{-37}\) then which of the following must be true?
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
MUST KNOW FOR THE GMAT:
• Even roots from a negative number are undefined on the GMAT (as GMAT is dealing only with Real Numbers): \(\sqrt[{even}]{negative}=undefined\), for example, \(\sqrt{-25}=undefined\).
• Odd roots have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).
BACK TO THE ORIGINAL QUESTION:
As \(-2^5=-32\), then \(x\) must be a little bit less than -2, hence \(x=\sqrt[5]{-37} \approx -2.1 \lt -2\). Thus \(x^3 \approx (-2.1)^3 \approx -8.something \lt -8\), so option D must be true.
As for the other options:
A. \(\sqrt{-x}=\sqrt{-(-2.1)}=\sqrt{2.1} \lt 2\), \(\sqrt{-x} \gt 2\) is not true.
B. \(x \approx -2.1 \lt -2\), thus \(x \gt -2\) is also not true.
C. \(x^2 \approx (-2.1)^2=4.something \gt 4\), thus \(x^2 \lt 4\) is also not true.
Answer: D
General Discussion
24 Jul 2018, 07:57
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If \(x=\sqrt[5]{-37}\) then which of the following must be true?
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
Answer: D
\(x=\sqrt[5]{-37}\)
=> \(x^5 = -37\)
=> \((-x)^5 =37 \gt 32 = 2^5\)
=> \(-x \gt 2\)
=> \(x^3 \lt -8\)
A. \(\sqrt{-x} \gt 2\)
B. \(x \gt -2\)
C. \(x^2 \lt 4\)
D. \(x^3 \lt -8\)
E. \(x^4 \gt 32\)
Answer: D
\(x=\sqrt[5]{-37}\)
=> \(x^5 = -37\)
=> \((-x)^5 =37 \gt 32 = 2^5\)
=> \(-x \gt 2\)
=> \(x^3 \lt -8\)
