Questions: Rationalize the denominator and simplify. (3√11 + √3) / (√11 - √3)

Solution Steps

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{11} - \sqrt{3}\) is \(\sqrt{11} + \sqrt{3}\). This will eliminate the square roots in the denominator. Simplify the resulting expression.

To rationalize the denominator of the expression \(\frac{3\sqrt{11} + \sqrt{3}}{\sqrt{11} - \sqrt{3}}\), we first identify the conjugate of the denominator. The conjugate of \(\sqrt{11} - \sqrt{3}\) is \(\sqrt{11} + \sqrt{3}\).

Multiply both the numerator and the denominator by the conjugate \(\sqrt{11} + \sqrt{3}\):

\[ \frac{(3\sqrt{11} + \sqrt{3})(\sqrt{11} + \sqrt{3})}{(\sqrt{11} - \sqrt{3})(\sqrt{11} + \sqrt{3})} \]

The denominator becomes a difference of squares:

\[ (\sqrt{11})^2 - (\sqrt{3})^2 = 11 - 3 = 8 \]

Expand the numerator:

\[ (3\sqrt{11} + \sqrt{3})(\sqrt{11} + \sqrt{3}) = 3\sqrt{11} \cdot \sqrt{11} + 3\sqrt{11} \cdot \sqrt{3} + \sqrt{3} \cdot \sqrt{11} + \sqrt{3} \cdot \sqrt{3} \]

This simplifies to:

\[ 3 \times 11 + 3\sqrt{33} + \sqrt{33} + 3 = 33 + 4\sqrt{33} + 3 = 36 + 4\sqrt{33} \]

Combine the results from the numerator and denominator:

\[ \frac{36 + 4\sqrt{33}}{8} \]

Simplify by dividing each term in the numerator by 8:

\[ \frac{36}{8} + \frac{4\sqrt{33}}{8} = \frac{9}{2} + \frac{\sqrt{33}}{2} \]

Final Answer