Question

c \\(\frac{2\sqrt{3} - \sqrt{2}}{5\sqrt{2} + \sqrt{3}}\\)

Explanation:

To simplify the fraction \(\frac{2\sqrt{3} - \sqrt{2}}{5\sqrt{2} + \sqrt{3}}\), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5\sqrt{2} + \sqrt{3}\) is \(5\sqrt{2} - \sqrt{3}\).

Step 1: Multiply numerator and denominator by the conjugate

$$ \frac{(2\sqrt{3} - \sqrt{2})(5\sqrt{2} - \sqrt{3})}{(5\sqrt{2} + \sqrt{3})(5\sqrt{2} - \sqrt{3})} $$

Step 2: Expand the numerator

Using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

Step 3: Expand the denominator

Using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\):

$$
LATEXBLOCK1
$$

Step 4: Combine the results

Putting the expanded numerator and denominator together, we get:

$$ \frac{11\sqrt{6} - 16}{47} $$

Answer:

\(\frac{11\sqrt{6} - 16}{47}\)