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 take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{49}{16}} \)
\( \frac{\sqrt{49}}{\sqrt{16}} \)
\( \frac{\sqrt{7^2}}{\sqrt{4^2}} \)
\( \frac{7}{4} \)
1\(\frac{3}{4}\)

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

\( \frac{18\sqrt{63}}{6\sqrt{9}} \)
\( \frac{18}{6} \) \( \sqrt{\frac{63}{9}} \)
3 \( \sqrt{7} \)

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:

36,000 fans x \( \frac{3}{4} \) = \( \frac{108000}{4} \) = 27,000 fans.

First, solve for \( \left|z + 5\right| \):

\( \left|z + 5\right| \) + 5 = -6
\( \left|z + 5\right| \) = -6 - 5
\( \left|z + 5\right| \) = -11

The value inside the absolute value brackets can be either positive or negative so (z + 5) must equal - 11 or --11 for \( \left|z + 5\right| \) to equal -11:

So, z = 6 or z = -16.