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8 November, 08:16

Simplify 6/2√3-√6+√6/√3+√2 - 4√3/√6-√2

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  1. 8 November, 08:23
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    -sqrt (2) + 3 sqrt (3) - sqrt (6)

    Step-by-step explanation:

    Simplify the following:

    (6 sqrt (3)) / 2 - sqrt (6) + (sqrt (6)) / (sqrt (3)) + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    6/2 = (2*3) / 2 = 3:

    3 sqrt (3) - sqrt (6) + (sqrt (6)) / (sqrt (3)) + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    Rationalize the denominator. (sqrt (6)) / (sqrt (3)) = (sqrt (6)) / (sqrt (3)) * (sqrt (3)) / (sqrt (3)) = (sqrt (6) sqrt (3)) / 3:

    3 sqrt (3) - sqrt (6) + (sqrt (6) sqrt (3)) / 3 + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    sqrt (6) sqrt (3) = sqrt (6*3):

    3 sqrt (3) - sqrt (6) + (sqrt (6*3)) / 3 + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    6*3 = 18:

    3 sqrt (3) - sqrt (6) + (sqrt (18)) / 3 + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    sqrt (18) = sqrt (2*3^2) = 3 sqrt (2):

    3 sqrt (3) - sqrt (6) + (3 sqrt (2)) / 3 + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    (3 sqrt (2)) / 3 = 3/3*sqrt (2) = sqrt (2):

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - (4 sqrt (3)) / (sqrt (6)) - sqrt (2)

    Rationalize the denominator. (-4 sqrt (3)) / (sqrt (6)) = (-4 sqrt (3)) / (sqrt (6)) * (sqrt (6)) / (sqrt (6)) = (-4 sqrt (3) sqrt (6)) / 6:

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) + (-4 sqrt (3) sqrt (6)) / 6 - sqrt (2)

    The gcd of - 4 and 6 is 2, so (-4 sqrt (3) sqrt (6)) / 6 = ((2 (-2)) sqrt (3) sqrt (6)) / (2*3) = 2/2 * (-2 sqrt (3) sqrt (6)) / 3 = (-2 sqrt (3) sqrt (6)) / 3:

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) + (-2 sqrt (3) sqrt (6)) / 3 - sqrt (2)

    sqrt (3) sqrt (6) = sqrt (3*6):

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2/3 sqrt (3*6) - sqrt (2)

    3*6 = 18:

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - (2 sqrt (18)) / 3 - sqrt (2)

    sqrt (18) = sqrt (2*3^2) = 3 sqrt (2):

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2/3 3 sqrt (2) - sqrt (2)

    Combine powers. (-2*3 sqrt (2)) / 3 = 2 sqrt (2) * 3^ (1 - 1) (-1):

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2 sqrt (2) * 3^ (1 - 1) - sqrt (2)

    1 - 1 = 0:

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2 sqrt (2) * 3^0 - sqrt (2)

    3^0 = 1:

    3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2 sqrt (2) * 1 - sqrt (2)

    Add like terms. 3 sqrt (3) - sqrt (6) + sqrt (2) + sqrt (2) - 2 sqrt (2) - sqrt (2) = - sqrt (2) + 3 sqrt (3) - sqrt (6):

    Answer: - sqrt (2) + 3 sqrt (3) - sqrt (6)
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