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Rationalize the denominator of the expression (2)/(3+sqrt(5)) and simplify.

Mathematics

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Answer

3−52\frac{3-\sqrt{5}}{2}

Step-by-step solution

6 steps

  1. Identify the expression to rationalize.

    The expression is 23+5\frac{2}{3+\sqrt{5}}.

  2. Multiply the numerator and denominator by the conjugate of the denominator.

    The conjugate of 3+53+\sqrt{5} is 3−53-\sqrt{5}.

    23+5×3−53−5\frac{2}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}

  3. Expand the numerator.

    2(3−5)=6−252(3-\sqrt{5}) = 6 - 2\sqrt{5}

  4. Expand the denominator using the difference of squares formula (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

    (3+5)(3−5)=32−(5)2=9−5=4(3+\sqrt{5})(3-\sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4

  5. Combine the simplified numerator and denominator.

    6−254\frac{6 - 2\sqrt{5}}{4}

  6. Simplify the fraction by dividing each term in the numerator by the denominator.

    64−254=32−52\frac{6}{4} - \frac{2\sqrt{5}}{4} = \frac{3}{2} - \frac{\sqrt{5}}{2}

    This can also be written as:

    3−52\frac{3-\sqrt{5}}{2}
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