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16 Sep 2014, 00:44
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
61% (01:43) correct
39%
(01:59)
wrong
based on 189
sessions
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16 Sep 2014, 00:44
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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:
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = \)
\(=\sqrt{7 + \sqrt{16*3}} - \sqrt{3}=\)
\(=\sqrt{7 + 4\sqrt{3}} - \sqrt{3}=\)
\(=\sqrt{4 + 4\sqrt{3} + 3} - \sqrt{3}=\)
\(=\sqrt{(2 + \sqrt{3})^2} - \sqrt{3} = \)
\(=(2 + \sqrt{3}) - \sqrt{3} = 2\).
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:
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = \)
\(=\sqrt{7 + \sqrt{16*3}} - \sqrt{3}=\)
\(=\sqrt{7 + 4\sqrt{3}} - \sqrt{3}=\)
\(=\sqrt{4 + 4\sqrt{3} + 3} - \sqrt{3}=\)
\(=\sqrt{(2 + \sqrt{3})^2} - \sqrt{3} = \)
\(=(2 + \sqrt{3}) - \sqrt{3} = 2\).
03 May 2015, 02:40
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Another smart way of doing this is like this:
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = X\)
\(\sqrt{7 + \sqrt{48}} = X + \sqrt{3}\)
\(7 + \sqrt{48} = 7 + 4*\sqrt{3} = (X^2+3) + 2*X*\sqrt{3}\)
Its not hard to figure out that X = 2
\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = X\)
\(\sqrt{7 + \sqrt{48}} = X + \sqrt{3}\)
\(7 + \sqrt{48} = 7 + 4*\sqrt{3} = (X^2+3) + 2*X*\sqrt{3}\)
Its not hard to figure out that X = 2
