Evaluating Radical Expressions

The question asks us to find the value of a mathematical expression involving square roots. The expression is given by:

\[ \left( \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \right) - \left( \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \right) \]

To evaluate this expression, we can simplify each fraction separately using a technique called rationalization of the denominator. Rationalizing involves multiplying the numerator and denominator by the conjugate of the denominator.

Step 1: Simplify the First Term

The first term is \( \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \). The conjugate of the denominator \( \sqrt{5} + \sqrt{3} \) is \( \sqrt{5} - \sqrt{3} \). We multiply both the numerator and the denominator by this conjugate:

\[ \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{(\sqrt{5} - \sqrt{3})^2}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

Now, we use the algebraic identities: \( (a-b)^2 = a^2 - 2ab + b^2 \) and \( (a+b)(a-b) = a^2 - b^2 \).

Numerator: \( (\sqrt{5} - \sqrt{3})^2 = (\sqrt{5})^2 - 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15} \)

Denominator: \( (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \)

So, the first term simplifies to:

\[ \frac{8 - 2\sqrt{15}}{2} = \frac{2(4 - \sqrt{15})}{2} = 4 - \sqrt{15} \]

Step 2: Simplify the Second Term

The second term is \( \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \). The conjugate of the denominator \( \sqrt{5} - \sqrt{3} \) is \( \sqrt{5} + \sqrt{3} \). We multiply both the numerator and the denominator by this conjugate:

\[ \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \]

Using the algebraic identities: \( (a+b)^2 = a^2 + 2ab + b^2 \) and \( (a-b)(a+b) = a^2 - b^2 \).

Numerator: \( (\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \)

Denominator: \( (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \)

So, the second term simplifies to:

\[ \frac{8 + 2\sqrt{15}}{2} = \frac{2(4 + \sqrt{15})}{2} = 4 + \sqrt{15} \]

Step 3: Subtract the Second Term from the First

The original expression is the difference between the first term and the second term:

\[ \left( 4 - \sqrt{15} \right) - \left( 4 + \sqrt{15} \right) \]

Remove the parentheses and simplify:

\[ 4 - \sqrt{15} - 4 - \sqrt{15} \]

Combine the like terms (constants and terms with \( \sqrt{15} \)):

\[ (4 - 4) + (-\sqrt{15} - \sqrt{15}) = 0 - 2\sqrt{15} = -2\sqrt{15} \]

The value of the expression is \( -2\sqrt{15} \).

Let's check this against the given options:

Our calculated value, \( -2\sqrt{15} \), matches Option 1.

Revision Table: Key Concepts Reviewed

Additional Information: Working with Radical Expressions

Working with radical expressions often requires applying properties of square roots and algebraic identities. Here are some important points:

• Product Property: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) (where \( a \ge 0, b \ge 0 \)). For example, \( \sqrt{15} = \sqrt{5 \times 3} = \sqrt{5} \times \sqrt{3} \).

• Quotient Property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) (where \( a \ge 0, b > 0 \)).

• Adding/Subtracting Radicals: You can only add or subtract radicals if they have the same radicand (the number under the root symbol). For example, \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \), but \( \sqrt{2} + \sqrt{3} \) cannot be simplified further.

• Rationalizing the Denominator: This is a standard technique to eliminate radicals from the denominator, making expressions easier to work with and compare. When the denominator is a binomial involving square roots (like \( \sqrt{a} + \sqrt{b} \) or \( a - \sqrt{b} \)), multiply by its conjugate.

Understanding these fundamental concepts is crucial for simplifying and evaluating expressions involving radicals.