Rationalizing the Denominator for Square Root Expressions

The problem asks us to simplify the expression \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\). To simplify this fraction involving square roots in the denominator, we use a technique called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

Understanding the Conjugate

The denominator is \(\sqrt{5}+\sqrt{3}\). Its conjugate is \(\sqrt{5}-\sqrt{3}\). Multiplying an expression by its conjugate helps eliminate the square roots from the denominator because it follows the difference of squares pattern: \((a+b)(a-b) = a^2 - b^2\).

Step-by-Step Simplification

We multiply the given expression by \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}\) (which is equal to 1, so it doesn't change the value of the expression):

Calculating the Numerator

The numerator is \((\sqrt{5}-\sqrt{3}) \times (\sqrt{5}-\sqrt{3})\). This is a perfect square: \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{5}\) and \(b = \sqrt{3}\).

• \(a^2 = (\sqrt{5})^2 = 5\)
• \(2ab = 2(\sqrt{5})(\sqrt{3}) = 2\sqrt{5 \times 3} = 2\sqrt{15}\)
• \(b^2 = (\sqrt{3})^2 = 3\)

So, the numerator becomes \(5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}\).

Calculating the Denominator

The denominator is \((\sqrt{5}+\sqrt{3}) \times (\sqrt{5}-\sqrt{3})\). Using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\):

• \(a^2 = (\sqrt{5})^2 = 5\)
• \(b^2 = (\sqrt{3})^2 = 3\)

So, the denominator becomes \(5 - 3 = 2\).

Final Result

Now, we combine the simplified numerator and denominator:

We can simplify this further by factoring out a 2 from the numerator:

Canceling the common factor of 2, we get:

Matching the Option

The simplified expression is \(4 - \sqrt{15}\), which corresponds to the second option provided.