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Difficulty:
85%
(hard)
Question Stats:
51% (01:52) correct
49%
(02:07)
wrong
based on 1810
sessions
History
Date
Time
Result
Not Attempted Yet
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = ?\)
A. 1
B. 1.7
C. 2
D. 2.4
E. 3
M11-18
A. 1
B. 1.7
C. 2
D. 2.4
E. 3
M11-18
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15 Mar 2014, 06:04
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Official Solution:
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = ?\)
A. 1
B. 1.7
C. 2
D. 2.4
E. 3
First, let's simplify the expression: \(\sqrt{7 + \sqrt{48}} - \sqrt{3} = \sqrt{7 + \sqrt{16*3}} - \sqrt{3} = \sqrt{7 + 4\sqrt{3}} - \sqrt{3}\). Observe that the answer choices are all rational numbers, so irrational parts in the expression must somehow cancel each other out. This observation should be a hint that we might be able to rewrite \(7 + 4\sqrt{3}\) in a way that makes it the square of something, which would allow us to extract the square root and simplify the expression. Bearing this hint in mind, we rewrite the expression as \(4 + 4\sqrt{3} + 3\). Upon examining the expression \(4 + 4\sqrt{3} + 3\), we notice that \(4 + 4\sqrt{3} + 3 = (2 + \sqrt{3})^2\).
Therefore:
Answer: C
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = ?\)
A. 1
B. 1.7
C. 2
D. 2.4
E. 3
First, let's simplify the expression: \(\sqrt{7 + \sqrt{48}} - \sqrt{3} = \sqrt{7 + \sqrt{16*3}} - \sqrt{3} = \sqrt{7 + 4\sqrt{3}} - \sqrt{3}\). Observe that the answer choices are all rational numbers, so irrational parts in the expression must somehow cancel each other out. This observation should be a hint that we might be able to rewrite \(7 + 4\sqrt{3}\) in a way that makes it the square of something, which would allow us to extract the square root and simplify the expression. Bearing this hint in mind, we rewrite the expression as \(4 + 4\sqrt{3} + 3\). Upon examining the expression \(4 + 4\sqrt{3} + 3\), we notice that \(4 + 4\sqrt{3} + 3 = (2 + \sqrt{3})^2\).
Therefore:
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = \)
\(=\sqrt{7 + \sqrt{16*3}} - \sqrt{3}=\)
\(=\sqrt{7 + 4\sqrt{3}} - \sqrt{3}=\)
\(=\sqrt{4 + 4\sqrt{3} + 3} - \sqrt{3}=\)
\(=\sqrt{(2 + \sqrt{3})^2} - \sqrt{3} = \)
\(=(2 + \sqrt{3}) - \sqrt{3} = 2\).
\(=\sqrt{7 + \sqrt{16*3}} - \sqrt{3}=\)
\(=\sqrt{7 + 4\sqrt{3}} - \sqrt{3}=\)
\(=\sqrt{4 + 4\sqrt{3} + 3} - \sqrt{3}=\)
\(=\sqrt{(2 + \sqrt{3})^2} - \sqrt{3} = \)
\(=(2 + \sqrt{3}) - \sqrt{3} = 2\).
Answer: C
16 Mar 2014, 20:59
Kudos
Bookmarks
If some1 is really fast in calculation then they might try this method as well.
\(\sqrt{ 7+ \sqrt{48}} - \sqrt{3}\) = \(\sqrt{ 14} - \sqrt{3}\) ( approximating 48 as 49)
= 3.7xxx - 1.732
which is close to 2.
but looking at the question it is better to do it the way Bunuel did it as GMAT doesn't want us to be good at calculations but at observation.
\(\sqrt{ 7+ \sqrt{48}} - \sqrt{3}\) = \(\sqrt{ 14} - \sqrt{3}\) ( approximating 48 as 49)
= 3.7xxx - 1.732
which is close to 2.
but looking at the question it is better to do it the way Bunuel did it as GMAT doesn't want us to be good at calculations but at observation.
