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Math Help - Quickie #3

  1. #1
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    Quickie #3

    Simplify: . \frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}

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  2. #2
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    Quote Originally Posted by Soroban View Post
    Simplify: . \frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}

    ] \frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}

    (4+\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}+\  frac{9\sqrt{6}}{2}...[a]

    (4-\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}-\frac{9\sqrt{6}}{2}...[b]

    (6+\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}+  \frac{13\sqrt{10}}{2}....[c]

    (6-\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}-\frac{13\sqrt{10}}{2}...[d]

    \frac{a+b}{c-d}=\frac{7\sqrt{10}}{13\sqrt{10}}=\boxed{\frac{7}{  13}}
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  3. #3
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    Quote Originally Posted by Soroban View Post
    Simplify: . \frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}

    Hello Soroban,,
    I really don't know why you call this problem a "Quickie" (I still have a crack in my head!)

    It took some time until I remembered the property:

    If x > y > 0 then

    \sqrt{x+y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}+\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}} or

    \sqrt{x-y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}-\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}}

    With these properties your term becomes:

    \frac{(4+\sqrt{15})^{\frac{3}{2}}+(4-\sqrt{15})^{\frac{3}{2}}} {(6+\sqrt{35})^{\frac{3}{2}}-(6-\sqrt{35})^{\frac{3}{2}}} =  \frac{(\sqrt{\frac{5}{2}}+\sqrt{\frac{3}{2}})^3+(\  sqrt{\frac{5}{2}}-\sqrt{\frac{3}{2}})^3}{(\sqrt{\frac{7}{2}}+\sqrt{\  frac{5}{2}})^3-(\sqrt{\frac{7}{2}}-\sqrt{\frac{5}{2}})^3}

    Expand the brackets and collect the roots with the same value. I've got:

    \frac{7 \cdot \sqrt{10}}{13 \cdot \sqrt{10}}=\frac{7}{13}

    Hapoooh!

    EB
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  4. #4
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    Lovely work, Galactus and EB!


    Don't know if the "Quickie" solution is any faster . . .


    We have: . \frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}


    Multiply top and bottom by 2^{\frac{3}{2}}\!:

    . . \frac{(8 + 2\sqrt{15})^{\frac{3}{2}} + (8 - 2\sqrt{15})^{\frac{3}{2}}} {(12 + 2\sqrt{35})^{\frac{3}{2}} - (12 + 2\sqrt{35})^{\frac{3}{2}}}

    . . =\;\frac{\left[(\sqrt{5}+\sqrt{3})^2\right]^{\frac{3}{2}} + \left[(\sqrt{5} - \sqrt{3})^2\right]^{\frac{3}{2}}} {\left[(\sqrt{7} + \sqrt{5})^2\right]^{\frac{3}{2}} - \left[(\sqrt{7} - \sqrt{5})^2\right]^{\frac{3}{2}} }

    . . = \;\frac{(\sqrt{5}+\sqrt{3})^3 + (\sqrt{5} - \sqrt{3})^3}{(\sqrt{7} + \sqrt{5})^3 - (\sqrt{7} - \sqrt{5})^3}

    . . =\:\frac{(5\sqrt{5} + 15\sqrt{3} + 9\sqrt{5} + 3\sqrt{3}) + (5\sqrt{5} - 15\sqrt{3} + 9\sqrt{5} - 3\sqrt{3})} {(7\sqrt{7} + 21\sqrt{5} + 15\sqrt{7} + 5\sqrt{5}) - (7\sqrt{7} - 21\sqrt{5} + 15\sqrt{7} - 5\sqrt{5}}

    . . = \:\frac{28\sqrt{5}}{52\sqrt{5}}

    . . =\:\boxed{\frac{7}{13}}

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