Question

what is the following quotient?

\\(\frac{1}{1+\sqrt{3}}\\)

• \\(\frac{\sqrt{3}}{4}\\)
• \\(\frac{1+\sqrt{3}}{4}\\)
• \\(\frac{1-\sqrt{3}}{4}\\)
• \\(\frac{-1+\sqrt{3}}{2}\\)

Explanation:

Multiply by the conjugate

Using the Radical Conjugates and Rationalizing the Denominator knowledge points

$$ \frac{1}{1+\sqrt{3}} = \frac{1 \cdot (1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} $$

Simplify the denominator

Using the Rationalizing the Denominator knowledge point

$$ (1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 $$

Simplify the fraction

Using the Algebraic Simplification knowledge point

$$ \frac{1-\sqrt{3}}{-2} = \frac{-(1-\sqrt{3})}{2} = \frac{-1+\sqrt{3}}{2} $$

Answer:

• (A) \(\frac{\sqrt{3}}{4}\)
• (B) \(\frac{1+\sqrt{3}}{4}\)
• (C) \(\frac{1-\sqrt{3}}{4}\)
• (D) \(\frac{-1+\sqrt{3}}{2}\) (Correct answer)