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# (√5+√5)(√5−√5)^2 =

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Math Expert
Joined: 02 Sep 2009
Posts: 53831

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23 Oct 2018, 23:59
00:00

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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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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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. $$20 - 2\sqrt{5}$$

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

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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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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.

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