The number of students taking German or Spanish is 12 + 15 = 27. Of that group of 27, 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 27 - 5 = 22 who are taking at least one language. 36 - 22 = 14 students who are not taking either language.

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

\( \frac{14\sqrt{8}}{2\sqrt{4}} \)
\( \frac{14}{2} \) \( \sqrt{\frac{8}{4}} \)
7 \( \sqrt{2} \)

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:

43,000 fans x \( \frac{3}{4} \) = \( \frac{129000}{4} \) = 32,250 fans.

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-7z^7}{8z^2} \)
\( \frac{-7}{8} \) z^{(7 - 2)}
-\(\frac{7}{8}\)z⁵