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

\( \frac{4}{8} \) x \( \frac{4}{9} \) = \( \frac{4 x 4}{8 x 9} \) = \( \frac{16}{72} \) = \(\frac{2}{9}\)

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 15 workers at the company now and that's enough to staff 5 crews so there are \( \frac{15}{5} \) = 3 workers on a crew. 9 crews are needed for the busy season which, at 3 workers per crew, means that the roofing company will need 9 x 3 = 27 total workers to staff the crews during the busy season. The company already employs 15 workers so they need to add 27 - 15 = 12 new staff for the busy season.

The number of students taking German or Spanish is 15 + 13 = 28. Of that group of 28, 7 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 28 - 7 = 21 who are taking at least one language. 27 - 21 = 6 students who are not taking either language.

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

6y⁶ x 3y⁶
(6 x 3)y^{(6 + 6)}
18y¹²

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{3 x 5}{9 x 5} \) - \( \frac{2 x 3}{15 x 3} \)

\( \frac{15}{45} \) - \( \frac{6}{45} \)

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

\( \frac{15 - 6}{45} \) = \( \frac{9}{45} \) = \(\frac{1}{5}\)