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# What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when

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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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10 Jun 2016, 13:23
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55% (hard)

Question Stats:

63% (01:28) correct 37% (01:27) wrong based on 325 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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What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when  [#permalink]

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10 Jun 2016, 15:27
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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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09 Jul 2017, 10:57
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$$(\sqrt{a + \sqrt{b}} + \sqrt{a - \sqrt{b}})^{2}$$ when $$a = 11$$ and $$b = 85$$?

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

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

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

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

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

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09 Jul 2017, 11:03
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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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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12 Jul 2017, 16:57
2
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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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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12 Aug 2017, 14:31
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)?
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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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10 Jan 2018, 22:44
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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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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10 Jan 2018, 23:09
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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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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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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26 Apr 2018, 06:19
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!!
Re: What is the value of ((a + b^(1/2))(1/2) + (a - b^(1/2))(1/2))^2, when &nbs [#permalink] 26 Apr 2018, 06:19
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