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Double square root

by neeti2711 » Thu Mar 30, 2017 12:37 am
What is the value of [√(9-√65) - √(9+√65)]^2?

(A) 10
(B) √65
(C) 18-5√13
(D) 18
(E) 18+5√13

OA: A

Please suggest using (a+b)^2 and then a^2-b^2.
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by GMATGuruNY » Thu Mar 30, 2017 2:32 am
neeti2711 wrote:What is the value of [√(9-√65) - √(9+√65)]^2?

(A) 10
(B) √65
(C) 18-5√13
(D) 18
(E) 18+5√13
BALLPARK.

[√(9-√65) - √(9+√65)]²

≈ [√(9-8) - √(9+8)]²

= (√1 - √17)²

≈ (1 - more than 4)²

= (less than -3)²

= more than 9.

The correct answer is A.
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by Brent@GMATPrepNow » Thu Mar 30, 2017 5:39 am
neeti2711 wrote:What is the value of [√(9-√65) - √(9+√65)]²?

(A) 10
(B) √65
(C) 18-5√13
(D) 18
(E) 18+5√13
Here's another approach:

We basically have the structure (x - y)²
(x - y)² = x² - 2xy + y², so let's use this to evaluate [√(9-√65) - √(9+√65)]²

[√(9-√65) - √(9+√65)]² = [√(9-√65)]² - 2[√(9-√65)][√(9+√65)] + [√(9+√65)]²
= 9 - √65 - 2√(81 - 65) + 9 + √65
= 18 - 2√16
= 18 - 2(4)
= 18 - 8
= 10

Answer: A

Aside: if you're wondering how I got 2√(81 - 65), notice that I used two rules:
rule #1: (√j)(√k) = √(jk)
rule #2: (a + b)(a - b) = a² - b²

So, 2[√(9-√65)][√(9+√65)] = 2√[(9-√65)(9+√65)] [by rule #1]
= 2√[9² - (√65)²] [by rule #2]
= 2√[81 - 65]

Cheers,
Brent
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