Rationalize the denominator. $\frac{11}{\sqrt{7} - \sqrt{3}}$

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{11}{\sqrt{7} - \sqrt{3}}\).
2. Recognize that the denominator is a binomial involving square roots, so multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(\sqrt{7} - \sqrt{3}\) is \(\sqrt{7} + \sqrt{3}\).
3. Multiply numerator and denominator by the conjugate: \(\frac{11}{\sqrt{7} - \sqrt{3}} \times \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}\).
4. Apply the difference of squares formula to the denominator: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = \sqrt{7}\) and \(b = \sqrt{3}\), so the denominator becomes \$7 - 3$.
5. Write the new expression as \(\frac{11(\sqrt{7} + \sqrt{3})}{7 - 3}\) and simplify the denominator to complete the rationalization process.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done to simplify the expression and make it easier to work with. For denominators with square roots, multiplying numerator and denominator by a suitable expression removes the radical.

Conjugates of Binomials

The conjugate of a binomial expression like (√7 − √3) is (√7 + √3). Multiplying a binomial by its conjugate uses the difference of squares formula, which eliminates the square roots in the denominator by producing a rational number.

Difference of Squares Formula

The difference of squares formula states that (a − b)(a + b) = a² − b². This property is used to simplify expressions involving conjugates, especially when rationalizing denominators containing square roots, by turning the product into a difference of perfect squares.