To rationalize a denominator of the form (√a - √b), we multiply both the numerator and the denominator by its conjugate. The conjugate of (√5 - √3) is (√5 + √3). This is because when we multiply a binomial by its conjugate, we use the identity (a-b)(a+b) = a² - b², which eliminates the square roots.

To rationalize a denominator of the form (√a - √b), we multiply both the numerator and the denominator by its conjugate. The conjugate of (√5 - √3) is (√5 + √3). This is because when we multiply a binomial by its conjugate, we use the identity (a-b)(a+b) = a² - b², which eliminates the square roots.

We multiply the given fraction by a fraction equivalent to 1, which is (√5 + √3)/(√5 + √3). This operation does not change the value of the original expression but helps in rationalizing the denominator.

We multiply the given fraction by a fraction equivalent to 1, which is (√5 + √3)/(√5 + √3). This operation does not change the value of the original expression but helps in rationalizing the denominator.

Using the algebraic identity (a-b)(a+b) = a² - b², we can simplify the denominator. Here, a = √5 and b = √3. So, (√5)² - (√3)² = 5 - 3 = 2. This eliminates the square roots from the denominator.

Using the algebraic identity (a-b)(a+b) = a² - b², we can simplify the denominator. Here, a = √5 and b = √3. So, (√5)² - (√3)² = 5 - 3 = 2. This eliminates the square roots from the denominator.

Rationalise the denominator: 1/(sqrt(5) - sqrt(3)).

Answer: $\frac{\sqrt{5} + \sqrt{3}}{2}$

Step-by-step solution

Step 1: Identify the conjugate

To rationalize a denominator of the form $(\sqrt{a} - \sqrt{b})$ , we multiply both the numerator and the denominator by its conjugate. The conjugate of $(\sqrt{5} - \sqrt{3})$ is $(\sqrt{5} + \sqrt{3})$ . This is because when we multiply a binomial by its conjugate, we use the identity $(a-b)(a+b) = a^2 - b^2$ , which eliminates the square roots.

Step 2: Multiply by the conjugate

We multiply the given fraction by a fraction equivalent to 1, which is $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ . This operation does not change the value of the original expression but helps in rationalizing the denominator.

Step 3: Simplify the numerator

Multiplying the numerator by 1 simply results in the conjugate itself, which is $(\sqrt{5} + \sqrt{3})$ .

Step 4: Simplify the denominator

Using the algebraic identity $(a-b)(a+b) = a^2 - b^2$ , we can simplify the denominator. Here, $a = \sqrt{5}$ and $b = \sqrt{3}$ . So, $(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2$ . This eliminates the square roots from the denominator.

Step 5: Write the rationalized expression

By combining the simplified numerator and denominator, we get the final rationalized expression. The denominator is now a rational number (2).

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