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605-655 (Medium)|   Algebra|   Exponents|      
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AbdurRakib
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 10 Jun 2016, 13:23
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C
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45% (medium)

Question Stats:

68% (02:10) correct 32% (02:26) wrong based on 1218 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
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C
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Bunuel
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 10 Jun 2016, 15:27
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AbdurRakib
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
Show more

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

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

Answer: C.
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 09 Jul 2017, 10:57
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\((\sqrt{a + \sqrt{b}} + \sqrt{a - \sqrt{b}})^{2}\) when \(a = 11\) and \(b = 85\)?


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


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


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


\(22 + 2 \sqrt{36} = 22 + 12 = 34\)


Hence, Answer is C
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 09 Jul 2017, 11:03
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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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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 12 Jul 2017, 16:57
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Bunuel
AbdurRakib
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
Show more


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

Answer: C
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 12 Aug 2017, 14:31
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AbdurRakib
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

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

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

Answer: C.
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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)?
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 10 Jan 2018, 22:44
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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.
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 10 Jan 2018, 23:09
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mrdlee23
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.
Show more

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 13 Jan 2018, 10:49
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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.
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 26 Apr 2018, 06:19
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destinyawaits
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.
Show more


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!!
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 11 Mar 2019, 05:57
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Bunuel
AbdurRakib
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
Show more


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.
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 24 Apr 2019, 00:23
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destinyawaits
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.
Show more


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!
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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when 19 Jan 2026, 09:25
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Noticing algebraic identities is key in questions like these
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