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(√5+√5)(√5−√5)^2 =

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(√5+√5)(√5−√5)^2 =  [#permalink]

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New post 23 Oct 2018, 23:59
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

57% (01:55) correct 43% (02:40) wrong based on 97 sessions

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\((\sqrt{5 + \sqrt{5}} - \sqrt{5 - \sqrt{5}})^2 =\)


A. \(10 - 4\sqrt{5}\)

B. \(10 - 2\sqrt{5}\)

C. \(20- 8\sqrt{5}\)

D. \(20 - 4\sqrt{5}\)

E. \(20 - 2\sqrt{5}\)

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Re: (√5+√5)(√5−√5)^2 =  [#permalink]

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New post 24 Oct 2018, 00:17
1
\((a-b)^2\) = \(a^2 + b^2 - 2ab\)

\((\sqrt{5 + \sqrt{5}})^2 + (\sqrt{5 - \sqrt{5}})^2 - 2 (\sqrt{5 + \sqrt{5}}) (\sqrt{5 - \sqrt{5}})\)

\(5 + \sqrt{5} + 5 - \sqrt{5} - 2 (\sqrt{25 - 5})\)

\(10 - 2(\sqrt{20})\)

\(10 - 4\sqrt{5}\)

OPTION : A
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Re: (√5+√5)(√5−√5)^2 =  [#permalink]

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New post 24 Oct 2018, 00:18
Bunuel wrote:
\((\sqrt{5 + \sqrt{5}} - \sqrt{5 - \sqrt{5}})^2 =\)


A. \(10 - 4\sqrt{5}\)

B. \(10 - 2\sqrt{5}\)

C. \(20- 8\sqrt{5}\)

D. \(20 - 4\sqrt{5}\)

E. \(20 - 2\sqrt{5}\)


\((\sqrt{5 + \sqrt{5}} - \sqrt{5 - \sqrt{5}})^2 =(5+√5)+(5-√5)-2*\sqrt{(5+√5)(5-√5)} = 10-2*\sqrt{(25-5)} = 10-4√5\)

Answer: Option A
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(√5+√5)(√5−√5)^2 =  [#permalink]

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New post 24 Oct 2018, 00:43
1
Bunuel wrote:
\((\sqrt{5 + \sqrt{5}} - \sqrt{5 - \sqrt{5}})^2 =\)


A. \(10 - 4\sqrt{5}\)

B. \(10 - 2\sqrt{5}\)

C. \(20- 8\sqrt{5}\)

D. \(20 - 4\sqrt{5}\)

E. \(20 - 2\sqrt{5}\)


Apply,
1) \((a-b)^2=a^2-2ab+b^2\)
2) \((x+y)(x-y)=x^2-y^2\)
3) \(\sqrt{m}*\sqrt{n}=\sqrt{m*n}\)

\((\sqrt{5 + \sqrt{5}} - \sqrt{5 - \sqrt{5}})^2\)=\(5+\sqrt{5}+2*(\sqrt{5 + \sqrt{5}})*(\sqrt{5 - \sqrt{5}})+5-\sqrt{5}=10-2*\sqrt{{(5+\sqrt{5})*(5-\sqrt{5})}\)=

\(10-2*\sqrt{5^2-(\sqrt{5})^2}\)=\(10-2*\sqrt{25-5}=10-2\sqrt{20}=10-4\sqrt{5}\)

Ans. (A)
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(√5+√5)(√5−√5)^2 =  [#permalink]

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New post 24 Oct 2018, 04:06

Solution


Given:
    • We are given an expression, \([\sqrt{(5 + √5)} - \sqrt{(5 - √5)} ]^2\)

To find:
    • The value of the given expression

Approach and Working:
    • \([\sqrt{(5 + √5)} - \sqrt{(5 - √5)}]^2 = [\sqrt{(5 + √5)}]^2 + [\sqrt{(5 - √5)}]^2 – 2 * \sqrt{(5 + √5)} * \sqrt{(5 - √5)}\)

      That is = \((5 + √5) + (5 - √5) – 2 * \sqrt{(5 + √5)} * \sqrt{(5 - √5)} = 10 – 2 * \sqrt{(5^2 – 5)} = 10 - 2√20 = 10 - 4√5\)
Hence, the correct answer is Option A.

Answer: A

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(√5+√5)(√5−√5)^2 =   [#permalink] 24 Oct 2018, 04:06
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