If $\frac {3 - \sqrt 3}{2 + \sqrt 3} = a + b\sqrt 3$, what are the values a and b?

1. Analyze the Question

We are given an expression involving surds: $\frac{3 - \sqrt{3}}{2 + \sqrt{3}}$. We need to simplify this expression into the form $a + b\sqrt{3}$ and identify the values of $a$ and $b$.

2. Explain the Concept

To simplify a fraction with a surd 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.

• The conjugate of $(2 + \sqrt{3})$ is $(2 - \sqrt{3})$.
• Multiplying by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ is essentially multiplying by 1, so the value of the expression remains unchanged.

3. Show the Solution

Step 1: Set up the multiplication

Step 2: Expand the numerator $(3 - \sqrt{3})(2 - \sqrt{3}) = 3(2) - 3(\sqrt{3}) - \sqrt{3}(2) + (\sqrt{3})(\sqrt{3})$ $= 6 - 3\sqrt{3} - 2\sqrt{3} + 3$ $= (6 + 3) + (-3\sqrt{3} - 2\sqrt{3})$ $= 9 - 5\sqrt{3}$

Step 3: Expand the denominator $(2 + \sqrt{3})(2 - \sqrt{3})$ This follows the difference of two squares pattern: $(x + y)(x - y) = x^2 - y^2$. $= 2^2 - (\sqrt{3})^2$ $= 4 - 3$ $= 1$

Step 4: Combine the results

Step 5: Compare with $a + b\sqrt{3}$ Comparing $9 - 5\sqrt{3}$ with $a + b\sqrt{3}$: $a = 9$ $b = -5$

4. Identify the Answer

The values are $a = 9$ and $b = -5$. This corresponds to option A.

5. Exam Tip

When rationalizing, always remember that the denominator will always result in a rational number (no square roots). If you still have a square root in your denominator after multiplying by the conjugate, check your signs—you likely made a mistake in the expansion!