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:

32,000 fans x \( \frac{5}{6} \) = \( \frac{160000}{6} \) = 26,667 fans.

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so subtract the coefficients and retain the base and exponent:

6a⁶ - 3a⁶
(6 - 3)a⁶
3a⁶

First, solve for \( \left|x - 3\right| \):

\( \left|x - 3\right| \) + 7 = 1
\( \left|x - 3\right| \) = 1 - 7
\( \left|x - 3\right| \) = -6

The value inside the absolute value brackets can be either positive or negative so (x - 3) must equal - 6 or --6 for \( \left|x - 3\right| \) to equal -6:

So, x = 9 or x = -3.

In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 16 workers at the company now and that's enough to staff 4 crews so there are \( \frac{16}{4} \) = 4 workers on a crew. 9 crews are needed for the busy season which, at 4 workers per crew, means that the roofing company will need 9 x 4 = 36 total workers to staff the crews during the busy season. The company already employs 16 workers so they need to add 36 - 16 = 20 new staff for the busy season.

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 11 are [11, 22, 33, 44, 55, 66, 77, 88, 99]. The first few multiples they share are [33, 66, 99] making 33 the smallest multiple 3 and 11 share.

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

\( \frac{6 x 11}{3 x 11} \) - \( \frac{2 x 3}{11 x 3} \)

\( \frac{66}{33} \) - \( \frac{6}{33} \)

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

\( \frac{66 - 6}{33} \) = \( \frac{60}{33} \) = 1\(\frac{9}{11}\)