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Multiply the fraction by 2-sqrt(3)2-sqrt(3).

Mathematics

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Answer

2−32-\sqrt{3}

Step-by-step solution

4 steps

To rationalize the expression 12+3\frac{1}{2+\sqrt{3}}, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 2+32+\sqrt{3} is 2−32-\sqrt{3}.

  1. Multiply the fraction by 2−32−3\frac{2-\sqrt{3}}{2-\sqrt{3}}.

    12+3×2−32−3\frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}

  2. Perform the multiplication.

    The numerator becomes 1×(2−3)=2−31 \times (2-\sqrt{3}) = 2-\sqrt{3}.
    The denominator is in the form (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2, where a=2a=2 and b=3b=\sqrt{3}.

    (2+3)(2−3)=22−(3)2(2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2

  3. Simplify the denominator.

    22−(3)2=4−3=12^2 - (\sqrt{3})^2 = 4 - 3 = 1

  4. Write the simplified fraction.

    2−31=2−3\frac{2-\sqrt{3}}{1} = 2-\sqrt{3}
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