Problem 54 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression with a rationalized denominator is \( \frac{11(\sqrt{7} + \sqrt{3})}{4} \).

Step 1. Recognize the Difference

Recognize the denominator as the difference of two quantities, expressed as \( a - b \) where \( a = \sqrt{7} \) and \( b = \sqrt{3} \).

Step 2. Multiply by conjugate

To eliminate the square roots, multiply both the numerator and the denominator by the conjugate of the denominator which is \( \sqrt{7} + \sqrt{3} \). This gives \( \frac{11(\sqrt{7} + \sqrt{3})}{(\sqrt{7} - \sqrt{3})*(\sqrt{7} + \sqrt{3})} \).

Step 3. Simplify denominator

In the denominator, apply the identity \( (a+b)(a-b) = a^2 - b^2 \). So the denominator simplifies to \( (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4 \). During this step, keep the numerator unchanged.

Step 4. Simplify the whole fraction

After dividing both the numerator and the denominator by the common factor 4, the whole fraction simplifies to \( \frac{11(\sqrt{7} + \sqrt{3})}{4} \). It is important to note that a fraction can always be simplified by identifying and eliminating common factors from the numerator and the denominator.