To simplify a radical, factor out the perfect squares:

\( \sqrt{75} \)
\( \sqrt{25 \times 3} \)
\( \sqrt{5^2 \times 3} \)
5\( \sqrt{3} \)

To divide fractions, invert the second fraction and then multiply:

\( \frac{3}{8} \) ÷ \( \frac{4}{9} \) = \( \frac{3}{8} \) x \( \frac{9}{4} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{3}{8} \) x \( \frac{9}{4} \) = \( \frac{3 x 9}{8 x 4} \) = \( \frac{27}{32} \) = \(\frac{27}{32}\)

The number of students taking German or Spanish is 5 + 12 = 17. Of that group of 17, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 17 - 5 = 12 who are taking at least one language. 20 - 12 = 8 students who are not taking either language.

A ratio of 5:1 means that there are 5 home fans for every one visiting fan. So, of every 6 fans, 5 are home fans and \( \frac{5}{6} \) of every fan in the stadium is a home fan:

39,000 fans x \( \frac{5}{6} \) = \( \frac{195000}{6} \) = 32,500 fans.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50]. The first few multiples they share are [15, 30, 45, 60, 75] making 15 the smallest multiple 3 and 5 share.

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

\( \frac{7 x 5}{3 x 5} \) - \( \frac{3 x 3}{5 x 3} \)

\( \frac{35}{15} \) - \( \frac{9}{15} \)

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

\( \frac{35 - 9}{15} \) = \( \frac{26}{15} \) = 1\(\frac{11}{15}\)