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

Answer: C

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