Rationalize the denominator and simplify: \( \frac{x}{x+\sqrt{3}} \) \( \frac{\sqrt{3}}{3} \) \( \frac{x \sqrt{3}}{3} \) \( \frac{x^{2}-x \sqrt{3}}{x^{2}-3} \) \( \frac{x^{2}-\sqrt{3}}{x^{2}-3} \)

To rationalize the denominator of \( \frac{x}{x+\sqrt{3}} \), you multiply both the numerator and the denominator by the conjugate of the denominator, which is \( x - \sqrt{3} \). So, you have: \[ \frac{x(x - \sqrt{3})}{(x+\sqrt{3})(x-\sqrt{3})} = \frac{x^2 - x\sqrt{3}}{x^2 - 3} \] This results in the simplified expression: \(\frac{x^2 - x\sqrt{3}}{x^2 - 3}\). This technique allows you to eliminate the square root from the denominator and express the fraction in a simpler form. Moreover, rationalizing the denominator can often make calculations easier, especially if you need to perform further operations with the resulting expression. This can be helpful in algebra, calculus, and beyond when dealing with various functions or equations!