Simplify: \frac{1}{\sqrt{5}-\sqrt{7}+\sqrt{12}}

Question

Simplify: \frac{1}{\sqrt{5}-\sqrt{7}+\sqrt{12}}

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Xavia 4 years 2021-08-21T17:11:21+00:00 2 Answers 10 views 0

Answers ( )

  1. Ladonna
    0
    2021-08-21T17:12:49+00:00
    Reply

    Answer:

    Multiply the numerator and denominator by the conjugate.

    Exact Form:

    1√5−√7+2√3

    Decimal Form:

    0.75267290…

  2. thienthanh
    0
    2021-08-21T17:13:07+00:00
    Reply

    Make use of the difference of squares identity,

    a^2-b^2=(a-b)(a+b)

    Let a=\sqrt5-\sqrt7 and b=\sqrt{12}. Then

    \dfrac1{(\sqrt5-\sqrt7)+\sqrt{12}}\cdot\dfrac{(\sqrt5-\sqrt7)-\sqrt{12}}{(\sqrt5-\sqrt7)-\sqrt{12}}=\dfrac{(\sqrt5-\sqrt7)-\sqrt{12}}{(\sqrt5-\sqrt7)^2-(\sqrt{12})^2}=\dfrac{\sqrt5-\sqrt7-\sqrt{12}}{5-2\sqrt{35}+7-12}=-\dfrac{\sqrt5-\sqrt7-\sqrt{12}}{2\sqrt{35}}

    Now multiply the numerator and denominator by √(35):

    -\dfrac{\sqrt5-\sqrt7-\sqrt{12}}{2\sqrt{35}}\cdot\dfrac{\sqrt{35}}{\sqrt{35}}=-\dfrac{(\sqrt5-\sqrt7-\sqrt{12})\sqrt{35}}{70}=-\dfrac{5\sqrt7-7\sqrt5-2\sqrt{105}}{70}

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