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GmatKiss GMAT Titan Default Avatar
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Please help! Post Sun May 06, 2012 12:21 pm
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    aneesh.kg GMAT Destroyer!
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    Post Sun May 06, 2012 12:30 pm
    The expression resembles this form:
    (a - b)^2
    where
    a = (7 + (29)^0.5)^0.5
    b = (7 - (29)^0.5)^0.5

    (a - b)^2
    a^2 + b^2 - 2ab
    a^2 = 7 + (29)^0.5
    b^2 = 7 - (29)^0.5
    2ab = 2*[(7 + (29)^0.5)(7 - (29)^0.5)]^0.5
    = 2*[20]^0.5
    = 4*[5]^0.5

    So (a - b)^2
    = 7 + 7 - 4[5]^0.5
    = 14 - 4[5]^0.5

    (C) is the answer

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    Last edited by aneesh.kg on Sun May 06, 2012 12:53 pm; edited 1 time in total

    GmatKiss GMAT Titan Default Avatar
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    Post Sun May 06, 2012 12:47 pm
    aneesh.kg wrote:
    The expression resembles this form:
    (a - b)^2
    where
    a = (7 + (29)^0.5)
    b = (7 - (29)^0.5)

    (a - b)^2
    a^2 + b^2 - 2ab
    a^2 = 7 + (29)^0.5
    b^2 = 7 - (29)^0.5
    2ab = 2*(7 + (29)^0.5)(7 - (29)^0.5)
    = 2*(49 - 29) = 40

    So (a - b)^2
    = 7 + 7 - 40
    = - 26

    (A) is the answer
    Wrong! You missed the whole root!!!

    aneesh.kg GMAT Destroyer!
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    Post Sun May 06, 2012 12:54 pm
    Ah, Yes I did. Thanks.
    Corrected the post above.

    Corrected Solution:

    The expression resembles this form:
    (a - b)^2
    where
    a = (7 + (29)^0.5)^0.5
    b = (7 - (29)^0.5)^0.5

    (a - b)^2
    a^2 + b^2 - 2ab
    a^2 = 7 + (29)^0.5
    b^2 = 7 - (29)^0.5
    2ab = 2*[(7 + (29)^0.5)(7 - (29)^0.5)]^0.5
    = 2*[20]^0.5
    = 4*[5]^0.5

    So (a - b)^2
    = 7 + 7 - 4[5]^0.5
    = 14 - 4[5]^0.5

    (C) is the answer

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    Post Sun May 06, 2012 1:39 pm
    Quote:
    What is the value of [√(7+√29) - √(7-√29)]²?

    -26

    2√29

    14-4√5

    14

    14+4√5
    The square of a value must be nonnegative.
    Eliminate A.

    Ballpark.
    [√(7+√29) - √(7-√29)]²

    ≈ [√(7+6) - √(7-6)]² Since √25=5 and √36=6, √29 is between 5 and 6.

    ≈ (√13 - 1)²

    ≈ 3² Since √9=3 and √16=4, √13 is between 3 and 4.

    ≈ 9.

    Since we overestimated the value of √13-1, the correct answer must be less than 9.

    Only answer choice C works:
    14-4√5 ≈ 14-8 ≈ 6.

    The correct answer is C.

    If we couldn't ballpark because the answer choices were closer in value, we could use the following approach.

    Remembering that (a-b)² = a²-2ab+b², use substitution:
    Let x=7 and y=√29.
    Substituting x=7 and y=√29 into [√(7+√29) - √(7-√29)]², we get:

    (√(x+y) - √(x-y))²

    = (x+y) - 2√((x+y)(x-y)) + (x-y)

    = 2x - 2√(x²-y²)

    = 2*7 - 2√(7² - 29)

    = 14 - 2√20

    = 14 - 4√5.

    The correct answer is C.

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