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Director  B
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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Question Stats: 66% (02:13) correct 34% (02:25) wrong based on 317 sessions

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What is the value of $$(\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}})^2$$ when a = 11 and b = 85?

A. 0

B. 22

C. 34

D. $$22+ 2\sqrt{85}$$

E. 242
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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AbdurRakib wrote:
What is the value of $$(\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}})^2$$ when a = 11 and b = 85?

A. 0

B. 22

C. 34

D. 22+ $$2\sqrt{85}$$

E. 242

$$(\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}})^2=(a+\sqrt{b})+2*\sqrt{a+\sqrt{b}}*\sqrt{a-\sqrt{b}} + (a-\sqrt{b})=$$

$$=2a+2\sqrt{a^2-b}=2*11+2*\sqrt{121-85}=22+2*6=34$$

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Re: What is the value of [m](\sqrt{a + \sqrt{b}} + \sqrt{a - \sqrt{b}})^{  [#permalink]

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$$(\sqrt{a + \sqrt{b}} + \sqrt{a - \sqrt{b}})^{2}$$ when $$a = 11$$ and $$b = 85$$?

$$(a + \sqrt{b}) + 2 (\sqrt{a + \sqrt{b}} * \sqrt{a - \sqrt{b}}) + (a - \sqrt{b})$$

$$2a + 2 \sqrt{(a + \sqrt{b})(a - \sqrt{b})}$$

$$2a + 2 \sqrt{(a^{2} - b)}$$

$$22 + 2 \sqrt{36} = 22 + 12 = 34$$

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##### General Discussion
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What is the value of [m](\sqrt{a + \sqrt{b}} + \sqrt{a - \sqrt{b}})^{  [#permalink]

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1
The expression is of the form (x+y)^2 = x^2 + y^2 + 2*x*y
where x =$$(\sqrt{a + \sqrt{b}})$$ and y = $$(\sqrt{a - \sqrt{b}})$$

Here, the expression become $$a + \sqrt{b} + a - \sqrt{b}$$ + 2*$$(\sqrt{a + \sqrt{b}})$$*$$(\sqrt{a - \sqrt{b}})$$

=$$2a + 2\sqrt{(a + \sqrt{b})(a - \sqrt{b})}$$ because $$\sqrt{a}*\sqrt{b} = \sqrt{a*b}$$

= $$2a + 2(a^2 - b)$$ because $$(x+y)(x-y) = x^2 - y^2$$

Substituting values,
The expression becomes $$2*11 + 2*\sqrt{121-85} = 22 + 2*\sqrt{36} = 22 + 2*6 = 34$$(Option C)
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Target Test Prep Representative V
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Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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Bunuel wrote:
AbdurRakib wrote:
What is the value of $$(\sqrt{a+b^\frac{1}{2}}+\sqrt{a-b^\frac{1}{2}})^2$$ when a=11 and b=85?

A. 0

B. 22

C. 34

D. 22+ $$2\sqrt{85}$$

E. 242

We see that the given expression is in the form of (x + y)^2, which equals x^2 + y^2 + 2xy.

We can let x = √(a + √b) and y = √(a - √b); thus:

x^2 = [√(a + √b)]^2 = a + √b

y^2 = [√(a - √b)]^2 = a - √b

2xy = 2√(a + √b)√(a - √b)

2xy = 2√[(a + √b)(a - √b)]

2xy = 2√(a^2 - b)

Thus, x^2 + y^2 + 2xy equals:

a + √b + a - √b + 2√(a^2 - b)

2a + 2√(a^2 - b)

Substituting 11 for a and 85 for b, we have:

2(11) + 2√(11^2 - 85) = 22 + 2√(121 - 85) = 22 + 2√36 = 22 + 12 = 34

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Manager  B
Joined: 26 May 2013
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Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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Bunuel wrote:
AbdurRakib wrote:
What is the value of $$(\sqrt{a+b^\frac{1}{2}}+\sqrt{a-b^\frac{1}{2}})^2$$ when a=11 and b=85?

A. 0

B. 22

C. 34

D. 22+ $$2\sqrt{85}$$

E. 242

$$(\sqrt{a+b^\frac{1}{2}}+\sqrt{a-b^\frac{1}{2}})^2=(a+b^\frac{1}{2})+2*\sqrt{a+b^\frac{1}{2}}*\sqrt{a-b^\frac{1}{2}} + (a-b^\frac{1}{2})=$$

$$=2a+2\sqrt{a^2-b}=2*11+2*\sqrt{121-85}=22+2*6=34$$

when factoring 2xy, how come you distribute the radical across the entire identity; (x+y)(x-y) = x^2 - y^2 ?

i can't understand why we just don't take 2(a - b^1/2). why do we need to also take the square root of (a-b^1/2)?
Intern  S
Joined: 29 Jan 2017
Posts: 42
Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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I have the same question as person above me (ak). Can someone please explain?

I believe \sqrt{3} x \sqrt{3} = 3 so I don't understand why the same wouldn't apply to this example.
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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mrdlee23 wrote:
I have the same question as person above me (ak). Can someone please explain?

I believe \sqrt{3} x \sqrt{3} = 3 so I don't understand why the same wouldn't apply to this example.

Yes, $$\sqrt{3} * \sqrt{3} = 3$$ but do we have the same expressions under the square roots in $$2*\sqrt{a+\sqrt{b}}*\sqrt{a-\sqrt{b}}$$ ? NO. Under the first root we have $$a+\sqrt{b}$$ (with + sign) and under another we have $$a-\sqrt{b}$$ (with - sign).

So, one should do the way it's shown in the solution: $$2*\sqrt{a+\sqrt{b}}*\sqrt{a-\sqrt{b}}=2\sqrt{(a+\sqrt{b})(a-\sqrt{b})}=2\sqrt{a^2-b}$$ (by applying $$(x+y)(x-y)=x^2-y^2$$).

Hope it's clear.
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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1
Everyone here is a math wizard... but alas, I am not, therefore, I ballparked the entire problem.

I estimated that the square root of 85 was 9 and that the square root of 20 and 2 were 4.5 and 1.4 respectively.

The resulting answer I got was ~36, which was close to 34.

Solved it in about 15 seconds. this method may be useful even if one understands the proper calculations involved.
Intern  B
Joined: 15 Dec 2017
Posts: 14
Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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destinyawaits wrote:
Everyone here is a math wizard... but alas, I am not, therefore, I ballparked the entire problem.

I estimated that the square root of 85 was 9 and that the square root of 20 and 2 were 4.5 and 1.4 respectively.

The resulting answer I got was ~36, which was close to 34.

Solved it in about 15 seconds. this method may be useful even if one understands the proper calculations involved.

I know it's best to solve the problems correctly, but this one really went over my head. I tried doing some "guess-timation" with the numbers like you suggested and it really helped. I'm also sadly not a math wizard, but it's nice to know I can use some kind of strategy to get by. Thanks!!
Intern  B
Joined: 14 Feb 2014
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GMAT 1: 660 Q45 V37 Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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ScottTargetTestPrep wrote:
Bunuel wrote:
AbdurRakib wrote:
What is the value of $$(\sqrt{a+b^\frac{1}{2}}+\sqrt{a-b^\frac{1}{2}})^2$$ when a=11 and b=85?

A. 0

B. 22

C. 34

D. 22+ $$2\sqrt{85}$$

E. 242

We see that the given expression is in the form of (x + y)^2, which equals x^2 + y^2 + 2xy.

We can let x = √(a + √b) and y = √(a - √b); thus:

x^2 = [√(a + √b)]^2 = a + √b

y^2 = [√(a - √b)]^2 = a - √b

What happens when you use the foil method for the above equation? I got stuck. I understand when you square a root the root sign drops. However I'm confused why this does not seem to work the same way when using foil.
Intern  B
Joined: 02 Apr 2019
Posts: 3
Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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destinyawaits wrote:
Everyone here is a math wizard... but alas, I am not, therefore, I ballparked the entire problem.

I estimated that the square root of 85 was 9 and that the square root of 20 and 2 were 4.5 and 1.4 respectively.

The resulting answer I got was ~36, which was close to 34.

Solved it in about 15 seconds. this method may be useful even if one understands the proper calculations involved.

In the same boat here but I don't really understand your logic...understand the estimating square root of 85 is 9, but unsure about the logic after. why are you trying to find the sqr root of 20 and 2? how did you arrive at 36? many thanks! Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when   [#permalink] 24 Apr 2019, 00:23
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