A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

33,000 fans x \( \frac{3}{4} \) = \( \frac{99000}{4} \) = 24,750 fans.

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

8\( \sqrt{3} \) x 4\( \sqrt{6} \)
(8 x 4)\( \sqrt{3 \times 6} \)
32\( \sqrt{18} \)

Now we need to simplify the radical:

32\( \sqrt{18} \)
32\( \sqrt{2 \times 9} \)
32\( \sqrt{2 \times 3^2} \)
(32)(3)\( \sqrt{2} \)
96\( \sqrt{2} \)

To simplify this fraction, first find the greatest common factor between them. The factors of 20 are [1, 2, 4, 5, 10, 20] and the factors of 60 are [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]. They share 6 factors [1, 2, 4, 5, 10, 20] making 20 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{20}{60} \) = \( \frac{\frac{20}{20}}{\frac{60}{20}} \) = \( \frac{1}{3} \)

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 share.

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

\( \frac{6 x 9}{5 x 9} \) + \( \frac{2 x 5}{9 x 5} \)

\( \frac{54}{45} \) + \( \frac{10}{45} \)

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

\( \frac{54 + 10}{45} \) = \( \frac{64}{45} \) = 1\(\frac{19}{45}\)