A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{4!}{6!} \)
\( \frac{4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5} \)
\( \frac{1}{30} \)

To multiply terms with radicals, multiply the coefficients and radicands separately:

2\( \sqrt{8} \) x 2\( \sqrt{4} \)
(2 x 2)\( \sqrt{8 \times 4} \)
4\( \sqrt{32} \)

Now we need to simplify the radical:

4\( \sqrt{32} \)
4\( \sqrt{2 \times 16} \)
4\( \sqrt{2 \times 4^2} \)
(4)(4)\( \sqrt{2} \)
16\( \sqrt{2} \)

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

36,000 fans x \( \frac{4}{5} \) = \( \frac{144000}{5} \) = 28,800 fans.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 2}{6 x 2} \) + \( \frac{2 x 1}{12 x 1} \)

\( \frac{16}{12} \) + \( \frac{2}{12} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{16 + 2}{12} \) = \( \frac{18}{12} \) = 1\(\frac{1}{2}\)