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):
\[

$$\begin{align*} &(2\sqrt{3})(5\sqrt{2}) + (2\sqrt{3})(-\sqrt{3}) + (-\sqrt{2})(5\sqrt{2}) + (-\sqrt{2})(-\sqrt{3})\\ &= 10\sqrt{6} - 2 \times 3 - 5 \times 2 + \sqrt{6}\\ &= 10\sqrt{6} - 6 - 10 + \sqrt{6}\\ &= (10\sqrt{6} + \sqrt{6}) + (-6 - 10)\\ &= 11\sqrt{6} - 16 \end{align*}$$

\]

Step 3: Expand the denominator

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

$$\begin{align*} &(5\sqrt{2})^2 - (\sqrt{3})^2\\ &= 25 \times 2 - 3\\ &= 50 - 3\\ &= 47 \end{align*}$$

\]

Step 4: Combine the results

Putting the expanded numerator and denominator together, we get:
\[
\frac{11\sqrt{6} - 16}{47}
\]

Answer:

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