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(7+43+7-43)2 is equal to which of the following?

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(7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 10 Jan 2019, 00:33
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A
B
C
D
E

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[Math Revolution GMAT math practice question]

\((\sqrt{7+4√3}+\sqrt{7-4√3})^2\) is equal to which of the following?

\(A. 32\)
\(B. 30\)
\(C. 24\)
\(D. 16\)
\(E. 12\)

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(7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post Updated on: 11 Jan 2019, 08:11
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

\((\sqrt{7+4√3}+\sqrt{7-4√3})^2\) is equal to which of the following?

\(A. 32\)
\(B. 30\)
\(C. 24\)
\(D. 16\)
\(E. 12\)


\((\sqrt{7+4√3}+\sqrt{7-4√3})^2\) ==> \(a^2+b^2+2ab\)

=\(7+4√3+7-4√3+2((\sqrt{7+4√3})(\sqrt{7-4√3})\)

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

=7+7+2(7^2-(√3*16)^2)

=14+2

=16

Hence D
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Originally posted by NandishSS on 10 Jan 2019, 01:52.
Last edited by NandishSS on 11 Jan 2019, 08:11, edited 1 time in total.
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Re: (7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 11 Jan 2019, 08:03
NandishSS wrote:

=7+7+2(7^2-√3*16)

=14+2



NandishSS Can you please explain how you reduced 2(7^2-√3*16) to 2?
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Re: (7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 11 Jan 2019, 08:11
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jedijodes wrote:
NandishSS wrote:

=7+7+2(7^2-√3*16)

=14+2



NandishSS Can you please explain how you reduced 2(7^2-√3*16) to 2?


(7^2-(√3*16)^2)

49-48=1

There was a typo so edited.
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(7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 11 Jan 2019, 09:10
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((7+4(3)^1/2)^1/2 + ((7-4(3)^1/2)^1/2)^2 is in the form (x+y)^2
Where x= (7+4(3)^1/2)^1/2 and y= (7-4(3)^1/2)^1/2
(x+y)^2 = x^2+y^2 +2(XY)
(7+4(3)^1/2)+(7-4(3)^1/2)+2((7+4(3)^1/2)(7-4(3)^1/2))
(4(3)^1/2) crosses out
Now we have 7+7+2(difference of squares)
Diff of squares =((7)^2-(4(3)^1/2)^2)^1/2) =(49-16(3))^1/2 =1
Now 7+7+2(1)=16 Ans is D

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Re: (7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 11 Jan 2019, 11:04
MathRevolution wrote:
[Math Revolution GMAT math practice question]

\((\sqrt{7+4√3}+\sqrt{7-4√3})^2\) is equal to which of the following?

\(A. 32\)
\(B. 30\)
\(C. 24\)
\(D. 16\)
\(E. 12\)


just a question of \((a+b)^2\) and a\(^2 - b ^2\) = (a-b) (a+b)

= 7+4√3+7-4√3 + 2 *\(\sqrt{7+4√3}\) * \(\sqrt{7-4√3}\)
= 14 + 2

Ans D.
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Re: (7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 11 Jan 2019, 11:10
((7+4(3)^1/2)^1/2 + ((7-4(3)^1/2)^1/2)^2
(a+b)^2= a^2+b^2+2*a*b
Thus (7+4(3)^1/2)+(7-4(3)^1/2)+2((7+4(3)^1/2)(7-4(3)^1/2
Diff of squares =((7)^2-(4(3)^1/2)^2)^1/2) =(49-16(3))^1/2 =1
Now 7+7+2(1)=16
Ans is D
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Re: (7+43+7-43)2 is equal to which of the following?  [#permalink]

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New post 13 Jan 2019, 18:03
=>

You should recall two properties about nested radicals:
\(\sqrt{a+b+2√ab}= √a+√b,\)
\(\sqrt{a+b-2√ab}= √a-√b,\) where \(a > b.\)

Together, these yield
\((\sqrt{7+4√3}+\sqrt{7-4√3})^2\)
\(= (\sqrt{7+2√12}+\sqrt{7-2√12})^2\)
\(= (√4+√3+√4-√3)^2\)
\(= (2√4)^2\)
\(= 16\)

Therefore, the answer is D.
Answer: D
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Re: (7+43+7-43)2 is equal to which of the following?   [#permalink] 13 Jan 2019, 18:03
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