If frac 3 - sqrt 3 2 sqrt 3 a b sqrt 3 what are... - JAMB Mathematics 2023 Question

To rationalize the given expression \(\frac {3 - \sqrt 3}{2 + \sqrt 3}\), we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac {3 - \sqrt 3}{2 + \sqrt 3} \times \frac {2 - \sqrt 3}{2 - \sqrt 3} \]

This simplifies to:

\[ \frac {(3 - \sqrt 3)(2 - \sqrt 3)}{(2 + \sqrt 3)(2 - \sqrt 3)} \]

Expanding the numerator and denominator:

Numerator:
\[ (3 - \sqrt 3)(2 - \sqrt 3) = 6 - 3\sqrt 3 - 2\sqrt 3 + 3 = 9 - 5\sqrt 3 \]

Denominator:
\[ (2 + \sqrt 3)(2 - \sqrt 3) = 4 - 3 = 1 \]

Now, the expression becomes:

\[ \frac {9 - 5\sqrt 3}{1} = 9 - 5\sqrt 3 \]

Comparing this with \(a + b\sqrt 3\), we get:
\[ a = 9 \]
\[ b = -5 \]

Therefore, the values are \(a = 9\) and \(b = -5\).