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Which of the following is equal to

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Math Revolution GMAT Instructor
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New post 11 Jan 2018, 01:49
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A
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E

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[GMAT math practice question]

Which of the following is equal to \((\sqrt{16+4√15}+\sqrt{16-4√15})^2\)

A. 10
B. 12
C. 24
D. 36
E. 40

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New post 11 Jan 2018, 03:18
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MathRevolution wrote:
[GMAT math practice question]

Which of the following is equal to \((\sqrt{16+4√15}+\sqrt{16-4√15})^2\)

A. 10
B. 12
C. 24
D. 36
E. 40


\((\sqrt{16+4√15}+\sqrt{16-4√15})^2\)
= \(16+4√15+16-4√15+2\sqrt{(16+4√15)(16-4√15)}\) (because \((a + b)^2 = a^2 + b^2 + 2*a*b\))
= 32 + 2\(\sqrt{256 - 16*15}\) (because \((a+b)(a-b)= a^2 - b^2\)) = \(32+8 = 40\)(Option E)
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Re: Which of the following is equal to  [#permalink]

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New post 11 Jan 2018, 05:19
\((a+b)^2\) = \(a^2\) +\(b^2\) +2*a*b

(\(\sqrt{16+4√15}\)+\(\sqrt{16−4√15}\))^2 = 16+4√15 + 16−4√15 +2* \(\sqrt{16+4√15}\) *\(\sqrt{16−4√15}\)

(a+b)(a-b) = \(a^2\)-\(b^2\)

=32 + 2* \(\sqrt{(16*16 - 4*4*15)}\)
=32 + 2* \(\sqrt{(16*(16 -1)5)}\)
=32+ 2*4
=40

Ans. E
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New post 14 Jan 2018, 18:32
=>

\((\sqrt{16+4√15}+\sqrt{16-4√15})^2\)
\(= (\sqrt{16+2√60}+\sqrt{16-2√60})^2\)
\(=(√10 +√6 + √10-√6))^2\)
\(=(2√10)^2\)
\(=4*10\)
\(= 40\)

Therefore, the answer is E.

Answer: E
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Re: Which of the following is equal to  [#permalink]

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New post 18 Jan 2018, 08:44
MathRevolution wrote:
[GMAT math practice question]

Which of the following is equal to \((\sqrt{16+4√15}+\sqrt{16-4√15})^2\)

A. 10
B. 12
C. 24
D. 36
E. 40


We can use the quadratic identity of (x + y)^2 = x^2 + y^2 + 2xy, in which:

√(16 + 4√15) = x and √(16 - 4√15) = y; thus:

x^2 = 16 + 4√15

y^2 = 16 - 4√15

2xy = 2√[(16 + 4√15)(16 - 4√15)]

Notice that (16 + 4√15)(16 - 4√15) is in the form of a difference of squares, and thus:

(16 + 4√15)((16 - 4√15) = 16^2 - (4√15)^2 = 256 - 240 = 16

So 2√(16 + 4√15)((16 - 4√15) = 2√16 = 2 x 4 = 8

Thus, the final value is 16 + 4√15 + 16 - 4√15 + 8 = 40.

Answer: E
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Re: Which of the following is equal to   [#permalink] 18 Jan 2018, 08:44
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