sjuniv32 wrote:
\(\sqrt{7-4\sqrt{3}}=\)
(A) \(2 - \sqrt{3}\)
(B) \(3 - 2\sqrt{3}\)
(C) \(5 -2\sqrt{3}\)
(D) \(\sqrt{7} - \sqrt{3}\)
(E) \(\sqrt{ 7} - 2\sqrt{3}\)
To be honest, my first instinct here was to see whether I could estimate. I couldn't remember off the top of my head how to simplify something like the original expression, and often, on the GMAT, if a problem only includes numbers - no variables or anything like that - you can at least eliminate several answer choices via estimation.
sqrt(3) is about 1.7, so 7-4sqrt(3) = 7-4(1.7) = 7 - 4 - 4(.7) = 7 - 4 - 2.8 = about 0.2. That's a very small value. Its square root will be a little bigger: I know that the square root of 0.25 is 0.5, so the square root of 0.2 should be around 0.4 or thereabouts. The right answer will definitely be less than 1/2.
Let's use that as a starting point with the answer choices.
(A) 2 - sqrt(3) is about 2-1.7, or 0.3. That seems close enough to keep it for now.
(B) 3 - 2sqrt(3) is actually negative. Eliminate B.
(C) 5 - 2sqrt(3) = 5-2(1.7) = 5-3.4 = 1.6. This is too big. Eliminate C.
(D) sqrt(7) - sqrt(3). Hmm, what's the square root of 7? I'm not sure, but I know it's got to be between 2 and 3. So, this could be close. I'll keep it.
(E) The previous answer was pretty close, so this one will probably be too small. Eliminate E.
Okay, all we're left with is (A) and (D). We want to know which one is the square root of 7 - 4sqrt(3). How can we check that, even if we aren't sure how to do the math in the question? Let's try squaring the two options we have left! The one that comes out to 7 - 4sqrt(3) will be correct.
(A) (2 - sqrt(3))^2 = 4 - 4sqrt(3) + 3 = 7 - 4sqrt(3)
There we go - we have a winner! The correct answer is (A). And even if you didn't want to estimate, you could just quickly square all five answer choices (and if you started with A, you'd be lucky and finish quickly!)
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