Simplify $\frac{ 5 + \sqrt{7}}{3 + \sqrt{7}}$

To simplify an expression involving a square root in the denominator, we use a process called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

1. Analyze the question

We are given: $\frac{5 + \sqrt{7}}{3 + \sqrt{7}}$ The conjugate of the denominator $(3 + \sqrt{7})$ is $(3 - \sqrt{7})$.

2. Explain the concept

When you multiply a binomial of the form $(a + \sqrt{b})$ by its conjugate $(a - \sqrt{b})$, you get a difference of two squares: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b$. This removes the square root from the denominator, making the expression easier to work with.

3. Show the solution

Multiply the numerator and denominator by $(3 - \sqrt{7})$:

Step 1: Expand the numerator $(5 + \sqrt{7})(3 - \sqrt{7}) = 5(3) - 5(\sqrt{7}) + 3(\sqrt{7}) - (\sqrt{7})(\sqrt{7})$ $= 15 - 5\sqrt{7} + 3\sqrt{7} - 7$ $= (15 - 7) + (-5\sqrt{7} + 3\sqrt{7})$ $= 8 - 2\sqrt{7}$

Step 2: Expand the denominator $(3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2$ $= 9 - 7$ $= 2$

Step 3: Combine and simplify

Divide both terms in the numerator by 2:

4. Identify the answer

Comparing our result to the given options, we find it matches option B.

5. Exam Tip

Whenever you see a surd in the denominator, don't panic! Always look for the conjugate. Remember that the conjugate is simply the same two terms with the middle sign flipped. If the denominator is $(a + \sqrt{b})$, multiply by $(a - \sqrt{b})$. If it is $(a - \sqrt{b})$, multiply by $(a + \sqrt{b})$.